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KAMN ,

c5T1A.a.oft exouµe on:

4· -p2+a2 2p2+a2. Onote exouµe 2 =

Me Pcicrri A.omov 't'flV 1tapancivco anooet;TI µta mea.vft anooet;TI yia ruxa.io opeoyrovio tpiycovo a.no tou� nuea.yopeiou� eivm TI e;ft�: 'Eotro to opeoyrovto tpiycovo K y E ABf µe opeft 't'fl ycovia. A. I:xriµa.t�ouµe to tetpciycovo LlEZH facrt roote Knee nA.eupci tou va. icroutm µe P+y , onco� oto 1ta. paKcitro oxftµa.. To tetpciycovo A.ot1tov LlEZH anoteA.eitm a.no N tfooepa. opeoyrovta. tpiycova. ioa µe to ABf Kaero� Km to te tpciycovo KAMN To eµpa.Mv tou tetpa.yrovou LlEZH iooutm /\ µe E (p+y)2• E1tiO'fl�, to eµpa.Mv tou tetpayrovou icroutm y µe to uepotoµa trov eµpa.orov trov teoocipcov iocov opeoycovicov tptyrovcov Km tOU tEtpa.yrovou KAMN' OTIAO.Oit z y B H V M .

4· -Py+a2 2Py+a2. 'Exouµe A.omov ott: 2 2 (p+y)2 2py+a (*). H npotacrri 11.4 tCOV Erozxeiwv, nou anooioetm OtoU� nuea.yopeiou�, eivm TI e ;ft�: (p+y)2 p2+y2+2py (**). Onote TI oxecrri (*) µe 't'fl Pofteeia 't'fl� oxecrri � (**) yivetm : (p+y)2 2py+(l2 <=> p2+y2+2py 2py+(l2 <=> p2+y2 (l2. I:ta. Erozxeia tou EutleioTI to 1tUea.yopeio eeropTlµa eivm TI npota.crri 1.47 Km TI oiamncocrft tou eivm TI e;ft�: Ev roz� opBoywvioz� -rpzydJvoz� w air6 V/� V/V opBf/v ywviav virowzvov<frf� irkvpa� -rGrpaywvov iE

EYKAEIAH:E B'

104 T.4/2

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Il't>Oayop&1&� Aiaopoµ.t�

oov t:OTi wu; an6 'l'WV 'fr/V op(hjv ywviav nt:pit:xov<JdJv nkvpdJv rerpaydJvou;.

LlrtA.a<>r, to eµpa.Mv tOU tetpayrovou µe nA.eupa trtV unoteivoucra Bf eivm icro µe to a8po1crµa tO)V eµpa orov tCOV tetpayrovcov µe nA.euper; ttr; Mo Ka9eter; nA.euper; toU tptyrovou. E>a anooei�ouµe A.omov on: ( BrE�) = (ABZH) +(ArKE>) . Anoo&�ll

<l>epouµe a.no to A trtv AA napaUriA.ri crtrtv BLl. Ta tpiycova ABLl Km BfZ eivm icra y1a.ti AB=BZ, BLl=Bf Km 01 ycovier; ABLl Km ZBf eivm icrer; crav a8po1crµa µtar; oper,r; Km trt<; ycoviar; B. Apa ta tpiycova ABLl Km BfZ eivm tcroouvaµa. fta to eµpaoov tou tptyrovou Bf Z exouµe: 1

(B rz) = -BZ·u . Dµcor; u = ZH apa z 2 (BrZ) =!BZ· ZH =!BZ2 =!(ABZH) (*) 2 2 2 fta to eµpaoov toU tptyrovou ABLl exouµe: (A B�) =).!_ B�· A�=.!_ (B�AM) (**) 2 2 Ano ttr; (*) Km (**) exouµe O'tl ABZH ( ) = ( B�AM) (1) Oµoicor; exouµe O'tl ta tpiycova BfK Km AfE eivm icra apa (BrK) = (Ar E) . Enicrri r;: fJ. A (BrK)=!rK ·KE>=!rK 2 =!(ArKE>) Km (ArE)=!rE·AE=!(rEAM) Apa exouµe on 2 2 2 2 2 E (ArKE>) = (rEAM) (2) t& Ilpocr9etouµe Kata µEA.rt nr; ( 1) Km (2): ABZ H) +(ArKE>) = ( B�AM) +(rEAM) = ( BrE�) . ( 1

Anoot1;11 TO'\) Bhaskara (lvoia, 12� auova�)

Me nA.eupa trtV unoteivoucra Br tou op9oycoviou tp1yc0vou ABf Ka.taO"Keuai;ouµe to tetpaycovo Bf LlE. Ano ttr; Kopuq>er; Ll Km E cpepouµe LlZl.Af Km EHl.LlZ. IlpoeKteivouµe trtV BA nou teµvet trtV EH crto 0. I:xriµmii;ovtm etcrt tfocrepa icra op9oyrovta tpiycova Km to tetpaycovo AZHE> µe nA.eupa icrri µe AZ = P- y , onote to eµpa.Mv tou eivm icro µe E2 = (p- y)2• Ta tfocrepa op9oyrovia tpiycova ABf, fLlZ, LlEH Km BEE> acpou eivm icra µeta�U tour; ea eivm lO'OOUvaµa Km to eµpaoov Ka9evor; eivm icro µe E1=!py . To eµpaoov tou tetpayrovou 2 2 BfLlE eivm icro µe E = a • Enicrri r; auto anoteA.eitm a.no ta tfocrepa op9oyrovia tpiycova Km to tetpaycovo AZHE>. Apa to eµpaoov tou eivm icro µe 1 E=4E1+E2 <=>a2 =4-py+(p- y)2 <=>a2 =P2+y2. 2 Anoot1;11 TOl> Leonardo Da Vinci (1490)

LtrtV KO.taO'KEUll tOU Euiliiori npocreerouµe O'tO crmµa to op8oyc0v10 tpiycovo LlEA (A=90°) icro µe to ABf (LlA=AB Km AE=Af) Km q>epouµe to tµr,µa E>H. <l>epouµe enicrri r; ta tµilµma KZ Km EA. To KZ 01epxetm a.no trtv Kopucpr, A tou tp1yc0vou ABf ytati 01 tpeir; ycovier; KAB, BAf Km fAZ exouv a9pomµa EYKAEIAHl: B' 104 T.4/3

• ' ' ' ' ' • ' • • • ' ' • • '

------- Ilt>Ouyop&wc; Amopoptc;

Ta tpiyrova ABf, AE>H Km �EA sivm icm µsta�l> wu<;. Ta t&tpcinl.zupa KE>HZ Km ABEA sivm icm yimi E>H=BE, KE>=AB, EA=HZ Km m yrovis<; KE>H=ABE = 90°+B, E>HZ=BEA = 90°+f . Apa ( Ke HZ) = (AB EA) ( ) Enicni<; 'tCl tstpcinl.zupa KBfZ Km Af �A &ivm icm ymti KB=�A, Bf=f �. fZ=Af Km 01 yrovis<; KBf=A�f = 90°+B, BfZ=�fA = 90°+f. Apa (K Brz ) = ( Ar 6 A) ( ) A no n<; ( *) KCll ( ** ) exouµs on ta &�ciyrova BfZHE>K KCll ABEA�f sivm icm KCll icroouvaµa. Av ano ta icm autci &�ciyrova, Cl<pmpfoouµs ta icra tpiyrova ABf, AE>H KCll �EA ea ncipouµs on (Br6 E) = (ABK0) +(ArZH) oriA.aoft a2 =P2+y2. Anoo&1;11 p& opma Tpiymva. Bhaskara (12� aui>vu� Ka& John Wallis (1685) r <l>epouµs to U\jfO<; A�. Ta tpiyrova AB� Km fBA sivm oµoia yimi sivm opeoyrovia Km 01 yrovis<; A1 Km f sivm icrs<; yimi sivm o�sis<; µs Kaests<; 180° (45°+90°+45°).

1tl.zupe<; (A�.lBf Km AB.lAf). Onots

AB 86 <:::> AB2 = B6. Br = Br AB

Oµoiro<; a1to ta oµom tpiyrova Af � Km BfA ea 1tcipouµs

't1'J

(1)

crxecni :

Ar r6 = <=>Ar2 = r6·Br (2) Br Ar (1)+(2) : AB2+AP =B6·Br+r&Br=(B6+r6}Br=Bf.Br=BP �riA.aoft p2+y2 = a2.

H snoµsvri a1toosi�ri &ivm tou James Abraham Garfield (1831-1881) 20°u 1tpoeopou trov HITA 1tou OOAOq>ovi}01'JK& crti<; 19 :Esntsµppiou 1881. H oriµocrisucni 'tl'J<; anOO&l�l'J<; EylV& to 1882, tva XPOVO µstci to eavmo tou, crto aµsptKaV\KO nsp1001Ko The mathematical Magazine. Anoo&1;11 TOl> James Garfield (1882) <l>epouµs 'tl'JV fE Kae&'tl'J CT'tl'JV Bf KCll rE = Br = a. Ano to B E cpepouµs E�.lAf. To tstpcinl.zupo ABE� &ivm tpanesio, aq>ol> AB 11 6E µs i>'l'o<; to M. Ta tpiyrova ABr KCll f �E &ivm icra ytati sivm opeoyrovm v £xouv Bf=rE=a KCll Ol yrovis<; BI = rl yimi sivm O�&is<; µs Kcl est&<; 1tAeupE<;. Apa �E=Af=P KCll r�=AB=y . To sµpaoov tou tpanssiou ABE� &ivm icro µs : ,

(ABE6) =_!. (A B +6E) ·A6=_!.(y+p)(p+y) =_!.(p+y)2(1) 2 2 2

To tpa1tesio anotsA&hm ano ta icra opeoyrovia tpiyrova ABf Km f �E µs sµpaoov E1 =.!.py Km 2 to opeoyrovio Km icrocrK&AE<; tpiyrovo BfE µs sµpaoov E =_!. a2 2 2

•

Apa to sµpaoov tou (ABE6)=2E1+E =.!.py+.!.a2 (2) 2 2 2 Ano tl<; (1) KCll (2) exouµs 1

-(P+ y)2 =-Pr+-a2 <=>p2+y2 = a2. 2 2 2 AnoO&\;'l TOl> P. Fabre (1888) Ms nl.zupci 'tl'JV unot&ivoucra Bf KCl'tacrK&ucisouµs to t&tpciyrovo Bf �E. <l>epvouµs to AH icro Kat napciAAl'JAO µs to BE. Tots ta tstpci1tl.zupa ABEH Km Af �H &ivm napaUriA6ypaµµa, svro ta tpiyrova ABf Km HE� sivm icra. Ano to f q>epouµs to fZ.lH�, A E oriA.aoft to U\jfO<; to'U Af �H. Ta Mo opeoyrovia tpiyrova ABf KCll f �z &ivm icra yiati exouv Bf=f � Kat Ol yrovis<; rI = r ymti 2 &ivm O�&is<; µ& Kci0&t&<; nl.zupe<;. Onot& r z = Ar = p . To sµpaoov to'U napaUriA.oypciµµou Af �H &i·

EYKAEIAHI: B'

104 T.4/4

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UvOayoptu:i; Amopo p.ti;

Vat i<ro µf: : (A r �H) = Af· rz = p2. Oµoiros to eµpaoov tOU 1tapaMT1Aoypaµµou ABEH eivm icJO µf: (ABEH) = y2. Apa : p2+y2 =(A r �H) +(A BEH) = (Ar�HEB) ( ) To e�ayrovo Ar�HEB eivm icroMvaµo µt to rerpayrovo Bf�E yiari ra rpiyrova ABr Km HE� eivm icm,o1tote cx.2 =(Ar�HEB) ( ) A1to ns(*)Km(**)£xouµe p2+y2 = cx.2. *

Anoitl�'l Tou P. Renan (1889)

Me 1tA.Eup€s ri,s AB Km Ar Kata<JKEUa�ouµe ra rerpayrova ABHZ Km Ar �E. IlpoeKteivouµt ro U'lfOs KA Kata rµi)µa AE>=Br Km crxeOia�OUµf: to tpiyrovo E>Br. Ta rpiyrova ABE> Km BrH eivm icra ytari £xouv AB=BH, AE>=Br Km 01 yroviEs BAE> Km HBf eivm icres yiari eivm aµpA.EiEs µe Ka0eres 1tA.Eup€s (AE>..LBr Km AB..LBH). ria ro eµpaoov rou rptyrovou ABE> £xouµe:

�

(ABE>) = AE>· BK = Bf· BK ,

e1ti<JT'ls

1 , 1 1 (BilI) =-BH· AB=-y2,o1tote: (ABE>)= -y2 (1) 2 2 2 K B r fta ro eµpaMv tou rptyrovou Are £xouµt: 1 1 2 , 1 1 1 � · Ar = -p , rris (Bf�) = -r ,o1tote: (ArE>) =-p2 (2). (ArE>) = -AE). fK = -Bf· fK , emc 2 2 2 2 2 Ilpocr0faouµt Kara µ€A.TI (1)+(2): (AB8) +(Ar 8) = .!.(p2+y2) ( ) 2 E1ti<JT'ls (ABE>) +(ArE>) =_!_Bf. BK +_!_Bf. Kr =_!_Bf. (BK +Kr) =_!_a2 ( ) 2 2 2 2 A1to tts (*) Km (**) £xouµt : .!.(p2+y2)= .!.a2 <=>P2+y2 = cx.2. 2 2 KAf:ivouµe e8ro ti,s a1to8ei�ets tOU 1tV0ayopeiou 0eropi)µaros. Na <JTlµf:lcO<JOUµf: on avaµecra crns a1t08ei�ets 1tOU 1tapa0€tet o Loomis crro PtPA.io rou 8cv U1tapxe1 Kaµia rp1yrovoµtrp1Ki},e1tet8i} 01 pamKoi tU1tOl rris tptyrovoµerpias crrrip�ovrm crto 1tV0ayopeto 0eropTlµa. D1tros xapaKrriptcrttKa ypaq>et "Trigo nometry is because the Pythagorean Theorem is", 8T1A.a8i) "H Tptyrovoµtrpia uq>icrrarm, e1tet8i} to Ilu0a yopeto E>effipT1µa uq>i<rrarm". E>a 8ouµe rropa ro avTi<rrpoq>o rou 1tV0ayopeiou 0eropi)µaros. :Era Ewzxda eivm TI 1tporacrri 1.48 Km TI 8ta'tU1trocri} rris eivm TI e�i)s: *

Eav rpzychvov w an6 µza� rcvv nJ...evpchv rcrpaycvvov iuov � wz� an6 rcvv A.oznchv wv rpzychvov JiJo nkvpchv rcrpaychvoz�, 1/ nepzex6µev11 ycvvia vn6 rcvv A.oznchv wv rpzychvov JiJo nkvpchv opB� euriv.

�TIA.a8i) av cre €Ya rpiyrovo to rerpayrovo µms 1tA.Eupas £xe1 eµpaMv i<ro µe ro a0poicrµa rrov eµpa8rov rrov terpayrovrov µe 1tAEUPEs ns Mo aA.A.Es 1tAEuPEs tou tptyrovou,tote to tpiyrovo eivm op0oyrovto Km £xe1 rriv op0i) yrovia µera�u rrov Mo 1tA.Euprov. Anoitt;q

'Ecrtro ABr Eva tpiyrovo tEtoto rocrte to eµpaMv rou tetpayrovou µt 1tA.Eupa rriv Br va eivm i<ro µe a0poicrµa rrov eµpaorov rrov rerpayrovrov µt 1tA.Eup€s rts AB Km Ar,8T1A.a8i) cx.2 = p 2+y2 (1) E>a a1to8e�ouµt on TI yrovia BAr eivm op0i). cl>€pouµt rriv A�..LAf Km Jtaipvouµt M = AB = y. Tore ra re rpayrova µt 1tAEuPEs A� Km AB eivm i<ra. Av 1tpocr0foouµe cre aura tO tetpayrovo µe 1tAEupa tl'IV Af tote tO a0potcrµa t(l)V eµpa8cOV t©V terpayrovrov µf: 1tAEuPEs tls AB Km Ar eivm icro µe to a0potcrµa t(l)V B eµpa8rov rrov rerpayrovrov µe 1tAEuPEs ns A� Km Ar. ll. v v A E1tet8i} oµros TI yrovia rA� eivm op0i), to a0potcrµa rrov eµpa8rov rrov tetpayrovrov µe 1tAEuPEs ns EYKAEIAH�

B' 104 T.4/5

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UuOayopeu:i; Aiaopoµti;

A/1 Km Ar eivm icro µe 'to eµ�aoov wu 'te'tpayffivou µe nA.eup<i TIJV f /1 (7tUeay6pe10 eerop11µa), 011A.aM1 f� 2 = �2+y2(2) Ano n; ( 1) Km (2) exouµe 6n: f� = fB = a. Ta 'tpiyrova ABf Km A/if eivm iaa ytmi exouv 'tpet� 7tArupe� icre� µia 7tp0� µia, 'tl� Af KOtvit, AB = M an6 Ka'ta<JKeut1, Km fB = f� . Apa fAB = fA� = 90° . e

E1ttKTU<Jll Tot> Ilt>Oayopdot> 0eropfiµaToi; IlpoTa<Jll VI.31 (.Ewzxeia)

11ive'tm opeoyrovto 'tpiyrovo ABf (A= 90°) Km e�ro'teptK<i auwu KamaKeu<i�ouµe noA.uyrova 6µ01a µem�u wu� nou µta nA.eup<i wu� eivm Kotvit µe wu 'tptyffivou. T6'te 'to eµ�aoov 'tOU noA.uyffivou µe nA.eup<i TIJV uno'teivouaa eivm icro µe w aepomµa 'trov eµ�aoffiv 'tOOV noA.uyffivrov µe nA.eupe� n� Mo Kaee'te� nA.eupe�. fao axt1µa m opeoyffivta BfEZ, AfH0 Km ABK.A ei vm 6µota µem�U 'tOU�. 07tO't& ea a7tOOel�OUµe O'tl E1= E1 +E3.

H K B

A1tooei�11

<l>epouµe w U'lfO� A/1, on6'te m 'tpiyrova ABf Km !1BA eivm 6µoia µ&'ta�u wu�. Bf AB � 2 , Apa-=AB =B�·Bf. AB B� B� , E3 = AB2 = B�· Bf =, , BfEZ Km ABKA etvm oµoia (1 ) apa Ta op eoyrovia Bf Bf2 E1 Bf2 E f� E1ti<JT)� 'ta opeoyffivta BfEZ Km AfH0 eivm 6µota <ipa 2 = (2) E1 Bf E3 E2 B� f� Bf (1)+(2): -+=-+- = = 1 � E2+E3 = E1. E1 E1 Bf Bf Bf Av Kma<JKeucmouµe e�ro'teptK<i wu opeoyroviou 'tptyffivou Kavov1Kci noA.Uyrova µe ioio nA.t1eo� nA.eu pffiv 7t.X l<J07tArupa 'tpiyrova tl KaVOVlKU m:v't<iyrova, 07t00� <J'tO napaKU'tOO <JXtlµa, ea exouµe O'tl E1=E2 +E3. ,

Au't6 mx(>et aav n6pmµa TI'l� np6'ta<JT)� Vl .3 1, yimi 'ta KavovtK<i no A.Uyrova µe tOlO 1tAtle0� 7tAeUpcOV eivm 6µota µe'ta�U 'tOU� Km Kaee eva µnopei va µemaX'lµana'tei ae opeoyffivto nou eivm taeµ�aOtK6 µe au't6 (Ilp6'ta<JT) II. 14). Eni<JT)�, av e�ro'teptKtlta µe Otaµe'tpou� 't� 7tArupe� 't01) 'tptyffivou ea exouµe Km n<iA.t 6n E1= E1 +E3. .

EYKAEIAHI:

B' 104 T.4/6

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A1t00Et;'I

IIvOayopEiEi; Ataopopti;

l:uµcpcova µe tTJV npotami XII.2 tcov Ewzxdwv, "o A.6yo� tcov sµpaorov Mo Ki>tlcov Ti 11µurutlicov si vm icro� µs t0 A&yo tcov tetpayrovcov tcov omµttpcov wu�". Onots -2 = -(1) Km - = L. (2) a2 EI . a2 EI p2

p2 + 2

y (1)+(2): -2 +- = +- = = - = l � E2 +E3 = E1• 2 2 2 a a2 a a EI EI ea oouµe tropa µta npO-mmi ano tTJ Evvaywy� tOU Il6.1t1tOU 1tOU eivm yeviKEU<TI'J tOU nu9ayopeiou 9scopijµat0�. I:.vvaycoy'll, Ilpchaaq IV.4

E�co-rsptKO. wu -rpiyrovou ABf KatacrKeuat.;ouµs -ra napaU11Mypaµµa ABEZ Km AreK. IlposK-rsivouµe -r� nl..eupe� EZ Km eK nou -reµvov-rm crw A. ct>epouµs tTJV 119 A µieu9eia AA nou -reµvsi tTJV nl..eupO. Bf crt0 cniµeio N Km naipvouµs t0 -rµijµa NM=AA. KmacrKeuat.;ouµe t0 napaA.A.11Mypaµµo BfLlH -reww rocr-rs NM//fLll/BH. Tots yia -ra sµpaoa. -rcov -rpirov napaU11A.oyp6.µµcov icrxt>si on E1 = E2 + E3 . A1t00Et;'I

p /E 2

IlposK-reivouµs tTJV Llf nou -reµvsi tTJV eK crt0 P Km 'tTJV HB nou -reµvsi tTJV ZE crw T K, To napaU11Mypaµµo AfeK sivm icroMvaµo µs t0 AfPA ono-rs (AfPA)=E2 . To ,1tapaU11A.6ypaµµo ABEZ sivm icroMvaµo µe t0 ABTA o1tO'te (ABTA)=E3. ·-----------Ta napaMT)AOypaµµa AfPA Km fLlMN eivm lO'OOuvaµa a<pou " z T exouv icre� P O.crel� (A A = NM) .Km ppicrKOV'tm µe-ra�u 'tCOV nap<lA A.iJA.cov PLl Km AM. Ono-re E2 = (AfeK) = (AfPA) = (fLlMN) (1) Oµoico� E3 = (ABEZ) = (ABTA) = (BHMN) (2) I I I I I I I I .

.. ·

(1)+(2): E2 + E3 = E1• H npo-rami au'tij icrxt>si Km yta onoiooijno-rs napaU11Mypaµµo BfLlH (µs nl..eupa tTJV Bf)

apKei -ra -rµijµaµa-ra NM = AA . ea tleicrouµe tO 6.p9po au-ro µe tTJ Ota<JTOA1\ T01) XPOV01) ano tTJV elOlKTJ escopia tTJ� crxenKOtT)'ta�. A� uno9foouµe on exouµe Mo napatT)pTJ'tE�. 0 eva� eivm aKiVTJtO� Km XPOVOµE'tprovta� eva <pmvoµevo ppiO'Kel on au-ro A.aµpavel xropa O'e XPOVO t. 0 a.A.Ao� Ktvehm µe -raxt>trJ'ta u co� 1tp0� tOV 1tpc0t0 Km XPOvoµe-rprov-ra� tO iOto <pmvoµevo ppiO'Kel on aU'tO A.aµpO.vel xropa O'e XPOVO to. TO'te yta tOU� Mo XPOVOU� t onou c TJ mxt>trJ-ra wu <pco-ro� ( c 3 108 m Is). Am)OEt;tt

A� uno9foouµs on eva payovi KlVehm npo� ta Oe�tO. µs taxUtTJ'ta u. Ano tTJ eemi A O'tTJV opo<pij tOU payo VlOU �eKlV6. eva <pCO'tOVlO KlVOUµEVO K6.9s-ra npo� t0 nO.-rcoµa µs taxUtrJ'ta C, 01tOU C TJ taxUtT)'ta tOU <pCO'tO�. 'Eva� napatT)pTJ-riJ� Ilo 1tOU ppi O'Ke'tm µfoa O"tO payovt 9a aVttATJ<p9ei on tO <pCO'tOVlO 1tpOO'ID1t'tel O'tO <TI'JµEio B. Av o XPOVO� tTJ� KiVTJmi� wu <pro-

EYKAEIAHI:. B' 104 T.4/7

.. . . . . .

I !

-------

nuOayopt:u:i; Aiaopoµti;

toviou yux tov 1ta.paTT)PlltTt 110 sivm t0, tots yta to ouicrTT}µa AB exouµs ott AB = c·t0. 'Eva� aKiVlltO� 1tapaTT) PlltTt� 111 1tOU �pim<stm €�co a1to to �ayovt ea avttA.11cpesi Ott to <pCOtOVtO OUl ypacpst TT}V tpoxta Ar (yta.ti to tpevo Ktvsitm npo� ta os�ta) Km XPStasstm XPovo t. To otacrTT}µa Ar d vm icro µs A r = C·t SVcO yux to OtUOTT)µa Bf exouµs Ott B r = u . t yuxti to �ayovt KlVSitm µs taxj> TT}ta u. To nueayopsto esc0p11µa crto opeoyffivto tpiycovo ABf ea µa� officrst : AB2 +Br2 =Ar2�(c·t0)2 +(u·t)2 =(c·t)2 t . Av l.UaouµHl]V <�i<J COOl] ro<; � t ea KITTaA��OUµ£ <m] OJ(SGlj : t = '

�1-(:r

u < exouµs Ott to < t' 7tpayµa 7tOU 011µaivst Ott 0 XPOVO� yux tOV Ktvouµsvo na c pa.t11p11tft "tpexst" mo apya crs crxe011 µs toV aKiVlltO 7tapa't11Plltit, 011M18it 0 XPOVO� "oiac:ntw-rm". t0 l ,IS·t0• To l 1=t6 yia fta mxp«6styµa av u = tOts = <ipa t Acpou u < c�

�,

Fffi � � =

wv aKiVlltO napaTT)Plltit icroouvaµsi µs 0,86 Af:nta yta tov Ktvouµsvo. t t u Av t©pa u = ,9c , t&ts = ,9 , Opa t = � o = -o _ = 2 29·t0 c 436 92

1-0 0 '

To 1 Af:rrto yw. TOV aKiVllTO 1tClP«TTIP11Tll UJOOuvaµt:i µt 0,44 Af:nta 'YlCl TOV K\VOilµtvo!

u c tciva ITTO µ110EY apa to = t . Ot Mo XPOVOl Aot7tOV 7tpaKttKa dvm foot av Ot tax;l>TT}tS� dvm µtKp€� crs crx€01l µs 'tllV tax;l>'tll tO. tOU <pcoto�, 7tpayµa 7t0U cruµ�aivst OTT)V Kae11µsptvft µa� scoft. H Ota<poponoi11011 tcov Mo :xpovcov yivstm mcre11tit av ot tax;l>TT}ts� nA.11masouv TT}V tax;l>TT}ta tou cpcoto�. I:TT}v napanavco 8tanpayµatsucr11 XPllcrtµonotftcraµs to Af:yoµsvo "A;iroµa TO'O Einstein" TT}� st8tK'ft� escopia� TT}� crxsttKOTT}ta�, Ott 11 tax;l>TT}ta tOU <pcoto� dvm ioux yta oA.ou� tOU� napaTT)plltE� 7t0U �pim<o vtm navco crs aopavstaKa crucrtftµa.ta, 011A.a8ft 11 taxUTT}ta tou cpcoto� dvm 11 µsyaA.Utsp11 nou µnopsi va U1tap�st OTT) <pU<J11. Km yta tOU� Mo 7tapaTT)p11t€� Ilo Km 111 11 tax;l>TT}ta tOU <pcotoviou dvm c Km oxt c +u yux tOV aKi VlltO, onco� smtacrcrst 11 µllxavtKft wu Nsutcova. Eni<J11�, escopftcraµs TT}V tpoxta Ar su%ypaµµ11. I:TT}v 1tpayµattKOTT}ta autft dvm Kaµm'.>A.11, µs µs yaA.11 oµco� a.Ktiva KaµnuA.6'tllta� met8it 11 anocrta<J11 Af dvm µtKpft. 11paKttKa A.mnov 11 Kaµm'.>A.11 Af ei vm sueeia.. Be�ma av

11 tO.xUtllta u sivm 1tOAU µtKpit crs crxe011 µs 'tllV tax;l>TT}tO. tOU cpcot b�, tots to tlacrµa

BIBAIOfPACl>IA I:.ron}p11i; Xp. fKO'OVTovpai;, I'ewµccpud� L1za

Nfo Kmu..ocpopfo

I>ra Bt�A.toncoA.eia: 1. Kopqn6:r!]�, TTJA.210-3628492,

2. :Ea��ciA.a�, 3. Ila'tciKT]�, 4. IloA.rrnia 5. AvtKouA.a�,

Jpoµt�, "0€µa.ta fscoµetpia.� ano TT}V apxmoTT} ta co� tov 20° mffiva", Aeftva 20 1 5 2. K.E.Eil.EK., EVTckiJ17 "Ewixda , toµot I, II, III, Aeftva 200 1 . 3 . Kenneth W. Ford, 10.aaauaj Kai aiJyX,Pov17 <pvai "�' EKMcrst� r. 11vsuµa.ttKOU, Aei]va 1980 4. Fourrey E., Curiosites Geometriques, Paris, Vuibert, 1 938 5. Loomis Elisha Scott, The Pythagorean Proposi tion, Wasinghton, NCTM, 1 940 "

EYKAEIAHI:. B' 104 -r.4/8

IOTHPHI XP. rKOYNTOYBAI

rEwµEtplKE<;

�1aopoµ£c;

MIA AMIIMONOJ:HMANTH J:YNAPTHJ:H f

rIA THN APieMHJ:H TtlN eETIKJlN PHTnN

T aiJ...iaKo� At\>'Ttp1]�,

0 fKfopyK Kavtop XP110'lµonoi11cre tOV nivaKa (I) yta va anoBe�el on to crUVOAO o: tCOV avaycoycov tlacrµatcov eivm apieµi)crtµo. :EtT)v epyacria auti) ea napoucrtacrouµe µia aµqnµovocri)µaVtT) cruvaptT)<JT) f: o: N* XPllmµonotrovtac; tov nivaKa (I). -

lllNAKAl: I

TonoettT)<JT) tcov eettKrov p11trov p

o-+

cr crto

N* x N* 7

0ecopouµe tT)V apieµ11<JT) tCOV avaycoycov tlacr µatCOV ( oncoc; EKaVe Kat 0 Kavtop) Kata µi)Koc; tCOV Btaycovicov, navtote ave�aivovtac; ano aptcrtepa npoc; ta Bel;ta, Km µoA.tc; teA.etrocret 11 api0µ11<J11 tcov avaycoycov tlacrµatcov µiac; Btaycoviou, tT)<; 07COiac; 0 nprotoc; opoc; eiva.t _!_ Kat 0 T.eAeUtaio c; opoc; eivm 0 K K (o qmmKoc; apieµoc; K), 11 apieµ11<J11 ea <JUvexicrtei ano tT)V apxfi Tfl<; e1tOµevT); Bmycoviou Km ano to I

' -�'\ '

apxiKo IV\.acrµa

-- .

K+l

:EtOV 7ttVaKa (I) napatT)pOUµB, On tO aepotcrµa tCOV opcov Kaee tlacrµatoc; 7COU �pi<JKetat <Je µia

<JU-yKeKptµEvri Bmyrovio eivm crtaeepo, �· Jtavco <JtT) Biayrovio nou apx�et ano to tlacrµa _!_ , to K aepoicrµa auto eivm icro µe K+I. fta <JUvtoµia ea <JUµ�oA.icrouµe tT) Btayrovto 1t01) apx�el ano to tlcicrµa ..!_ µe K

XPll<Jtµonoti)crouµe <JtT) <JUVEXeta ym eUKoA.ia µac; tov nivaKa (II) avti tou nivaKa (I). EYKAEIAID: B ' 104 T.4/9

d( _!_)

Kat ea

----- MIA AMC>IMONOI:HMANTH I:YNAPTHI:H

f rIA THN API0MHI:H TON 0ETIIillN PHTilN -----

ni:\AKAl.: ll

Ot 0enKoi Pll'tOi

E>uµisouµe 6n µe TI'IV 1toUa1tA.acnacrtud1 cruvap'tTl<Jll <p tou Euler p picrKouµe to 1tA.fj0o� tcov 1tprotcov ° q 1tp0� 'tOV n (ne N ) 'tEtOlCOV rocrte: 1 �q�n. OpiSouµe 6n: <p(l)= l . 61tOU p A1t6 'tTl yvcocrtfj eecopia TI'I� <JUVclPTI'l<Jll� ( p ) p - p 1tpro't0� Kat Ke N°. (T2): cl>(n) n 1µe n = p�' · p�2 ... p�· (avclA.U<Jll tou 11=

{ ;J { :J ... ( :J

K-I

n cre yiv6µevo 1tprotcov 1tapay6vtcov) (T3) : <p( a P y . . . p)= <p(a) cp(p)cp(y) . . . <p(p ), av ot a, p , y ,. . .,p e N" Kat ava ouo eivm 1tpc0t01 µeta�u 'tOU�. I A1t6 'tOV (TI) yta p=2 KCll I<C2 =>cl>( 2K)= 2K - 2K- = apno� fta p;C2 =>cl>( PK)= PK - PK-I = apno� co� Ota<popa OUo 1teptt'tcOV fta n = p�' · p�2 ... p�· =>cl>( n) = c!>(P�') c!>(p�2) ... c!>(p�·) = aptto� co� ytv6µevo apncov. ·

t!l

•

EYKAEIAHI:. B' 104 't'.4/10

-----

MIA AMCl>IMONOl:HMANTH l:YNAPTHl:H f rIA THN API0MHl:H TnN 0ETIKnN PHTnN -----

I:uµrrspm1µa: 0 apt0µoi; <p(n) civm apnoi; yta Ka9e n EN* µe n>2. 'faov 7tivaKa II 7tapaTI)pouµe on:

:ETI)V �( ! ) U7tUPXEt 1 avayroyo KAacrµa µE a0potcrµa oprov tOV 2 Kut tcrxj>Et: <p(2)= 1 . 1

• •

:ETI)V �( .!. ) U1tUPXOUV 2 avayroya KAficrµata µe a9potcrµa OpOOV tOV 3 Kut tcrxj>Et: cp(3)=2 2

•

:ETI)V �( .!._) U7tUPXOUV 6 avayroya KAficrµata µE a9potcrµa OpOOV tOV 9 Kut tcrxj>Et <p(9)=6 8

:ETI)V �( _..!.._) U7tUPXOUV 1 0 avayroya KAacrµata µE a9potcrµa OpOOV tOV 1 1 Kut tcrxf>et: cp( l 1 )=10 10 'Etcrt o&mouµe9a va a7to&ei�ouµe on OTIJV � -1- u7tapxouv cp(v) avayroya tlacrµata, 7tou Ka9tva wui; v -1 EXEt a0potcrµa trov oprov tOU 'tOV V. fta 'tO OK07t0 auto, ea a7to&Ei�ouµe TI)V E7tOµEV117tPOta<J11: TIPOTA:EH l fta v�2, VEN* U1tUPXOUV cp(v) avayroya KA.acrµata, 7t0U Ka9tva wtO ama ExEt a9potcrµa trov oprov 'tOU 'tOV v. Arr6on�q: E>eropouµe tva apxtKO a7toKoµµa Tv-l (v�2) wu, &riA.a&i) Tv_1 = { 1 ,2,3, . . . ,v-2, v- 1 }. i. Ot 7tpc0t0t 7tpoi; 'tOV v ea civm KU7tOta crtotxeia tOU Tv- 1 7tA.i)0oui; cp(v) Km µaA.tcrta TI)V 7tpcOTI) 9e<J11 KatEXEt 0 1 Kut TI)V tEMUtaia 0 v- 1 . ii. Av a e Tv - l Km (a,v)=l tote Kut (v-a,v)=l µe wui; a Km v-a va a7texouv e�icrou a7to wui; 1 Kut v- 1 avticrtotxa acpou la- 1 l=lv-a-(v- 1 )I; &rtl..a&i) Ot 7tpcOtOt 7tpoi; 'tOV V eµcpavi�ovtm ava �euyri, etcrt rocrtE ta µEA.rt Ka9e �El>youi; va antxouv e�icrou ano wui; 1 Km v- 1 avticrtoixa Km va sxouv a0poicrµa v. Ilpayµan: a+(v-a)=v. iii. Av tropa crxriµaticrouµe ta tlacrµata (2) 7tOU eivm avayroyat ' ( 1 ) Kut ta tlacrµata v •

)

(

·,

_a_

v-a.

tote to 7tA.i)9oi; wui; eivm cp(v), acpou ot apt0µritsi; trov (1) eivm nprotot 7tpoi; tov v Km ot tou autou nA.i)9oui; (Myro wu ii) apt9µritsi; trov (2) eivm ot U7tOAot7tot 7tpcOtot npoi; wv v wu Tv-l . Ka0tva A.otnov ano ta tlacrµma ( 1 ) eivm < 1 Km Ka9tva a7to ta tlacrµma (2) eivm > 1 . Tiapaonyµa: 'Ecrtro v=l 0. Ilprowt npoi; tov 1 0 wu T 9 eivm ot apt9µoi 1 ,3, 7 ,9. :Euve7troi;, w crl>voA.o trov avayroyrov KAacrµatOOV { .!.,�,2.,2. } µe 7tArt9tKO apt0µo 4=cp( 1 0) avi)KOUV OTI)V � -1- =� -1- = 9 7 3 1 v-1 10-1

( ) ( )

( �J . 'Etcrt, av exouµe tva avayroyo tlacrµa � µnopouµe aµforoi; va opicrouµe TI) &tayrovto E1tl onoiai; �picrKEtm. Auti) eivm ri � ( ) . Ano ta npomouµeva enetm on: 7tpc0TI) &tayrovtoi; � ( T) 1tEPIBXEt l (=cp(2)) avayroya tlacrµata. &El>tEPTt &tayrovtoi; � ( ±) 7tEPIBXEt 2 (=cp(3)) avayroya tlacrµata. Apa ( ) I tpiTI) &tayrovtoi; � ( �) 7tEptEXEt 2 (=cp(4)) avayroya tlacrµata. Apa ( ) f I �-

TI)£

1 K+A-1

H H

r 3. =f(2)=1+2= f,q,(a.) a=2

r � =f{3)=1+2+2=

a=2

<!>(a.)

Apa, yta tov nA.rt9tKo apt9µo n tou cruvol..ou trov avayroyrov tlacrµatrov, nou Ka9tva ano auta ftatiav (v-a,v)=O� O/v Kat O/v-a� O/v-(v-a)=a=> O/(a,v)=l � O=!

t ftatiav (a, v-a)=o=> o/a Kat O/v-a => O/(v-a)+a=v

O/(a,v)=l => 0=1

EYKAEIAHI: B' 104 T.4/1 1

-----

MIA AMCl>IMONOI:HMANTH I:YNAPTHI:H f rIA THN API0MHI:H TfiN 0ETIKnN PHTfiN -----

K+ I

avi)Kel O''t� K npcine� 5myrovte� sxouµe: n=cp(2)+ cp(3)+ . . . +cp(K)+cp(K+ 1) it n =Lei>( a) KCll enet5it cp(l )=l n

ea sxouµe:

a=I

ea eivm:

K-

= f'.

( Cl) - l . Enet5it

f ( K)

K+ I

=Lcj>(a) -1

6µco� 'tO 'tel...eumio avaycoyo tlacrµa Tll� /),, (3) A1t6 'tOV W1t0 (3) naipvouµe:

a=I

eivm 'tO 'teAeu'taio avaycoyo tlacrµa Tll� f),,(

_K-1 1

E>ecopouµe 'tO avaycoyo tlacrµa

( _!_) K eivm 'tO l ' bTJA. 0 K, �

( - =i K

( Cl) - l '

a=I

K2'.2 01tOU 0

Ta avaycoya tlacrµa'ta 1tOU aµfoco� S1tOV'tCll 'tOU

, K-1 , 1 K , -etvCll 'ta -, ... ,- 1tOU aVT)KOUV O'TIJV e1tOµeVT) vlCl'YCOVlO 1 K 1 ,

a=2

A Ll

( -1 ) K

•

Tll� /),,( .!._ ) Kat Scr'tCO O'tl au't6 Ka'tsxet Til e esO'T) O'TIJV au�oucra 5m5oxfl K

1 O' K , 1 'tCOV -, ... , - , ... , - avaycoycov 1V1.acrµa'tcov TIJ� ( )

••

�

A Ll

e esO'TJ

fipocpaVOO� ea tcrx\>et:

r(:)= "�1<j>(a)-1 +e

( )= er p

f (K

- l) + 0 <::>

( )= i <j>( er p

a=I

Cl) - l + 0

, it e1tet5it cr+p=K+ 1 ea elVCll:

(4)

Dnco� 6µco� eA.SxeTJ (Ilp6'taO'TJ 1 cr'to

(iii))

o apieµTJ'tft� cr Tou

'tcOp<l 'tO

eivm nprow� npo� Tov K+ 1. E>ecopouµe

apxlKO a7t6Koµµa Tcr = { 1 ,2,3, .. . ,0'} 'tOU N*. Av avfll;TJ 'tftcrouµe TO crl>voA.o B Tcov crwixeicov wu Tcr• nou eivm nprowt npo� wv K+ 1 'tO'te: i.

ii.

motxeio 'tOU B ea eivm apteµT)'tft� avaycoyou tlacrµaw� Tll� !),,( .!._ ) (Ilp6'taO'T) 1, 'tO iii).

Knee

Enet5it

o cr eivm w Tel...euTaio crwixeio wu Tcr fae'tat 6n o nA.T)etK6� apieµo� Tcov crwixeicov Tou B

'taU't\l;e'tCll µ£ 'tOV e, 1tOU opiset TIJ esO'T) 'tOU avaycoyou tlacrµa'tO�

er p

O'TIJV /),,( .!._) K

ApKei A.otn6v va emA.Ucrouµe w en6µevo np6PA.TJµa: «Na ppeeei o 7tA.T)etK6� apieµ6� e 'tOU cruv6A.ou 'tCOV x e Tcr nou eivm Tswta rocr'te: (x, K+ 1)=l . Aua11

Ot x e Tcr µe (x,K+ 1)=1 6nou cr+ p = K+1 = p�' p�2 p:· eivm ot cpumKoi apteµoi nou 5ev 5tmpouv'tm an6 wu� PP p2, , Ps . 'Ecr'tco 6n Kanom crwixeia wu Tcr sxouv TIJV t5tOTIJ'ta I l • bTJA. va 5tmpouV'tm 5ta P l )))) )))) )) )) »» »» »» 12, bTJA. va btCllpOUV'tCll blCl P2 •••

•••

»»

18, 511A..

va 5tmpouV'tm 5m Ps Me A(lj), l �Ss cruµpoA.isouµe wv nA.T)etK6 apieµo wu cruv6A.ou Tcov crwixeicov wu T<J ' nou sxouv TIJV t5t6TIJ'ta � n.x. A(l2) eivm o nA.T)etK6� apieµo� Tou cruv6A.ou Tcov crwixeicov wu T0 nou 5tmpouvTm 5ta p2. Me A(lj,Ip) cruµpoA.isouµe 'tOV 1tAT)etK6 apteµ6 'tOU cruv6A.ou 'tCOV O''tOtxeicov 'tOU T O' 1tOU 'tO Kaeeva sxet 'tt� t5t6TIJ'te� Ij,Ip 5TJA.. va 5tmpouvTm 5m wu ytvoµevou PjPp µe l �, p:Ss Km Pj=fPp• tln. TsA.o� µe A(l1,12, ...,15) cruµpoA.ll;ouµe wv 1tATJ0tK6 apieµo wu cruv6A.ou Tcov cr'tmxeicov wu T0 nou w Kaeeva sxet 't� lblOTIJ'te� l1,l2,...,ls (5TJA.a5it va btCllpOUV'tCll bta 'tOU ytvoµevou P1P2 P ) ·

cruv5uacrnK6� rono� wu

µa� 5ivet wv e e N* nou eivm o

· · ·

e=cr- LA(Ii)+LA(li'IP)- LA(li'IP,lµ)+...+(-1)5 A(l1,I2,.••,I.) j,p,µ j j,p 7tA.T)etK6� apteµo� wu cruv6A.ou B 'tCOV crwixeicov wu T0 nou 3EV

Sylvester

EYKAEIAHI:. B ' 104 'T.4/12

-----

MIA AMCl>IMONOI:HMANTH I:YNAPTHI:H r rIA THN API0MHI:H TnN 0ETIKnN PHTnN -----

Kaveva ano toui:; P 1 ,P2 ,. · · ·Ps · Auto crri µaivet Ott Ka0e OT01xeio TOU B eivm evai:; cpucrtKO<; ap10µoi:; npcinoi:; npoi:; TOV K+ 1 (it o+p) Km avftKet oto T 'Etm yia to npoPA.Tlµa µai:; o npoTlrouµevoi:;

oimpo'6vrm µe

t. [ � ] + t.[ P ;�P.l - ;� [ P ; p� PJ · · + ( - 1)' [ p, p,� p.l 01tOU OTO 2° a0potoµa 1t.X. nftpaµe roi:; napovoµaotsi:; ta y1voµeva ava OUO foacpopettKOOV µtta�u TOU<; ano

M°' yivemi:

0•

9=a-

wui:; p 1 ,p2 , . . . ,ps µe oA.oui:; TOui:; ouvaTOui:; tponoui:; (cruvouaoµoui:;). :Eto 3° a0poicrµa · nftpaµe roi:; napovoµaotsi:; ta y1voµeva ava tptci:>v 01acpopettKci:>v µeta�u TOui:; ano wui:; P 1 ,P2 · · · .,ps µe oA.oui:; TOU<; ouvatoui:; tponoui:;, tln. 'Etcrt teA.tKa 0 mnoi:; (4) yivetm 0 napaKatro: mnoi:; (5)

()

"f"' IJ>( a ) a=I

1+0

-I [�Pi ] + Ii.P [ Pi · pp ] -Ii,p,µ [ Pi · pp · pµ ] + 0

i =I

(To cri>µpoA.o [x] eivm TO aKspmo µtpoi:; TOU x e R ) Ilapa&e1yµa: 'Eatro 11 <n>vapTl)(Jl) r : Q: --+ N * . Na pptOd 11 tiµ1]

Ai>aTI

cr = .!.!. . Apa o+p=l 1 + 3=14 Km 14 2 7 . Dµroi:; o+p- 1=13. · 3 p

( [ ] + [ ; ]) + [ ] = r

. . .

+ - 1 )5

( )

(

11 . 3

[ P1 · P 2 · · p. ] ( 0

. . .

Omm: ( ) t. r 131

+( a )- I + 1 1 -

11 2

1 2 .17

- 1 +1 1 - (5 + !) + O = 62

Ano TOv nivaKa (III).

IlapaTT1 Pftae1i:;

l. H 2.

cruvapTfl <JTt g : N· � N· µe mno: g(n) = I,<l>(a.) eivm µia ( 1 - 1 ) cruvapTfl crfl acpou yta n:;tn' (n,n e N*) 1crx;Ue1 g(n);tg(n ' ) Me XPitcrri Tfl<; 1 '1' napatftPTl <JTt <; Km tou mnou ( 5) Tfl<; f faetm aµforoi:; Ott Km TI f eivm µia ( 1 - 1 ) cruvapTfl crfl · Ai:; 0eropitoouµe tci:>pa TflV h, avtiotpocpTI Tfl<; cruvapTfl <JTt <; f, OT1A.aoit h : N • � Q: '

a=l

Il po PA.11 µa: Av h(30)= cr , Tott va IJptOd to avuyroyo KMiaµa � . p

Aua'l :

Apx1Ka ano TOV mvaKa III ppioKouµe no10 ano ta a0poioµata L cl>(a) eivm icro it TO aµtoroi:;

µtKpotepo tou 30. :ETflV nepintrocrft µai:;

f, cl>( a ) a=I

a=I

. EnoµSvroi:;, o+p- 1 = 9

�

cr + p = 1 0 = 2 · 5 onote to

avayroyo tlaoµa cr avftKet OTfl .::\( .!. ). AvttKa0toTOuµe OTOV tU1tO (4) Km sxouµe: 9 p f

( o) = P

Apa 3 cr

Dµroi:;,

( ) L I + TI 30=28- 1 +0 ( [ ; J + [ � J) + [ 1� J Km ene1oit o< 1 0 [ 1� J Km (6) [;] + [� ]

�- · L cl>(a ) - I

a=I

'1, f

cr -

[; J [ � J >O +

a=I

cl>( a ) -

�

0 =3 e�icrrocrri yivetm:

cr - 3 0 , onote o e {4,5,6,7,8,9} .

Me 0oKtµsi:; OTflV (6) ppicrKouµe ott autft aA.rt0el>e1 y1a cr=7 <::::> p=3. L\rtA.a<>it EYKAEIAIU: B ' 104 t.4/13

� p

'!._ 3

h(30).

M a8 11 µ aTI KO i a1 aywv1aµoi M a811 µ ar1 Kt� 0Au µrr1a6E�

E.M.E.

3411

EAA q v 1 Kq MaO q paT 1 Kq OAu1-111 108a 110 Apx1p1]011�" 4 Map't'io1> 2017

Qt;f!ll� f!lltl®l.�\i' ��t�V

npoPJ.riµa 1 . AiVtTat o;l>"(cOVlO Tpiyrovo ABC p t AB < AC c(O,R). 0 K6KM><; c1

( A,AC) Ttpvtt TOV K6tdo c(O,R) <YTO <n] pdo D Kat Tl]V npotKT«<n] TI]<;

n>.£upac; CB aTo <n]ptio E. Av 11 tl>8da AE Ttpvt1 Tov Ki>tdo c(O,R) aTo <n]ptio F Kat G d

.vm To <Yl>pptTptKo TOl> E roe; npoc; TO B, va anood;tTt oTt To TtTpan>.£upo FEDG dvm ty ypa'l/tpo.

A\J<nl ( 1 o; Tporro"): To tctpcinAf:upo AFBC civm eyycypaµµtvo crwv ri>tl.o i

• .

(c), cipa:

F; = ACB = C .

• I • • I •

j cifA,AC)

• . . • '

' ' ' '

' ' •

:ExiJµa To tpiycovo AEC civm icrocrKcMi; {0t AE Km AC civm aK-tivci; wu ri>tlou (c 1 )) cipa:

E 1 = AC B = C . Ano tti; tcrOt11tci; tcov ycovuov npori>ntct ott F1 = E 1 , onotc w tpiycovo BEF Eivm icrocrKcMi; Km ( ). BE=BF Kata cruvtncm: Ovoµci�ouµc cl = ECD x KCll ano tOV KUKAO (c 1 ) ea txouµc Ott EAD = 2x ' ( coi; cniKcvtp11), onotc EAB +BAD = 2x (2) (3 ) EmnMov, ano wv ri>tl.o (c) txouµc ott: BAD = C 1 = x Ano tti; (2) Km (3 ) txouµc ott EAB = BAD = x , onotc 11 AB civm �txotoµoi; crw tcrocrKcMi; tpiycovo EAD. :Euvcnroi; civm µccroKci0ctoi; t11i; ED, cipa BE=BD. (4) Ano tti; tcrOt11tci; ( 1 ) Km ( 4), Ka0roi; Km ano TllV npocpaviJ (/..6yco cruµµctpiai;) tcroTilta BE=BG , cruµncpaivouµc Ott BE = BF = BG = BD, onotc to tctpcinl..cupo DEFG civm cyypci\j/tµo crc ri> tlo µc Ktvtpo to B. 2 °; Tporro". To tctpcint..cupo AFBC civm cyycypaµµtvo crtov ri>tlo (c), cipa: F1 = AC B = C . To tpiycovo AEC civm tcrocrKcMi; (ot AE Km AC civm aKrivci; tou ri>tl.ou (c2)) cipa: E 1 = ACB C . Ano tti; tcrot11tci; tcov ycovtrov npori>ntct ott F; = E 1 , onotc w tpiycovo BEF Eivm tcrocrKcMi; Km B E = BF Kata cruvtncm: (5 ) . Ano w eyycypaµµtvo tctpcinAf:upo ABDC txouµc: I\ = ABC = B . Ano to tcrocrKcMi; tpiycovo =

EYKAEIAH� B ' 104 't.4/14

------

ADC sxouµt: 1\

Ma911pa'tiKo{ Aiayrovurpoi - Ma911pa'tiKt� 01..'ll pnulo� ------

ACD Ano tt<; ouo ttA.tutait<; tcrOtT)tE<; yrovirov, sxouµt Ott ACD B Km Kata cruvsntta cl B - c . Ano to tpiyrovo ABE sxouµt: Al =B El B - C ' onott: Al cl B - C Km tnttOT} Ot yrovit<; BD = BF Al ' cl tivm eyytypaµµEvt<; crtov Kl'.>tlo (c ), ea tcrx;Utt: (6). Ano tt<; icrotrJtt<; (5) Km (6), Ka0ro<; Km ano tTJV npoq>avf) p..oyro cruµµetpia<;) tcrOtT)ta BE=BG cruµntpaivouµe ott BE = BF = BG = BD, onott to tttpanAf:upo DEFG tivm tyypa\jftµo crt Kl'.> tlo µe Ktvtpo to B. =

rl pb fD.•uw 2 . 0t:ropoi>µt: a11µEio A

'TOt> E1tUtEOOt> Kat TpE� Et>9t:U:� 1t0l>

1tt:pvoi>v U1t0 Ul>'TO Kat zropi�Ot>V 'TO E1tl1tEOO GE

6 ToµEi�. I:t: Kcl9E Toµta 5 <J11 -

t>1tapzot>V <J'TO E<JCO'TEptKO 'TOt>

µEia. Yno9tTot>µt: O'Tt Ta 3 0 <J11 µEia

1t0t> fJpi<JKOV'Tat <J'TOt>�

TOµEi� Ei

Vat ava Tpia µ11 <Jt>VEt>Ot:taKa. Na anooEi;t:'TE O'Tl \)1tclPXOt>V 'TOt>A.axi

<J'TOV 1000 Tpiyrova µt KOpt> <p t� 'T(l

<J11 µEia «t>Ta (Trov

Toµtrov) TO o

noia nt:ptizot>v TO A EiTt mo taro TtptKo 'TOt>� Ei'Tt <J'Tl� 7t>..£t> pt� 'TOl>�. i\ i> cn1 :

IlapatrJpouµt ap:xtKa ott yia onoiaoftnott tnU..oyft 6 <Jflµtirov, tva ano Ka0t toµsa, OT}µioupytitm tva t;ayrovo (KUpto it µT} KUpto) to onoio ntpIB:;(tt to <Jflµdo A. Ano ta 6 auta <Jflµtia OT}µioupyou-

(�)

vtai

l:xf1µa 2

tpiyrova cr< m\voAo.

0a unoA.oyicrouµt nocra toUAa:;(t<:rtOV ano auta ntpis:xouv to <rflµdo A. Av S:;(OO OUO <Jflµtia ano Kata KOpUq>it toµti<;, tott napatrJpro ott sxro tm A.oyft yia tTJV tpitT) Kopuq>ft tOU tpt yrovou ano ouo toµti<;. fia napaotiy µa, yia ta <Jflµtia B, C tou napaKatro crxf1µatO<;, OnOtO <JflµEto Km napouµt ano tov KOKKtvo it tov npamvo toµSa S:;(OUµt tpiyrovo nou ntptS:;(Et to <Jfl µtio A. Ynap:xOUV 3 �tl>yT} Kata KOpuq>it tO µSrov, tnoµSvro<; Km £:xouµe 5 5 E1ttAo ys<; yia tTJ �a<Jfl BC Km Tl tpitTJ Kopu q>ft tm.Myttm µe 2 5 tponou<;. Eno µSvro<; E:;(OUµt <:rUVOAtKa tOUAa:;(tcrtOV 3 2 5 3 = 6 5 3 tt'toia tpiyrova nou 1tE pt£xouv to A. Av tropa sxro KOpUq>E<; crt evaua; ·

EYKAEIAHE B ' 104 T.4/15

------

Mu6qµuT\Koi Aiuyroviaµoi - Mu6qµuT\Kt; 0/..llµnull>t;

-------

wµti<; ( 01tCJ)<; q>CltVttm CrtO axitµa 4), tOtt 1tUAt txro tpiyrovo 1tOU 1ttPIBXtt 'tO a1iµtio A. Auta µnopti va tivm titt aav to CBD tht aav 'tO EFG. :Eav 'tO CBD unapxouv 5 . 5 . 5 = 5 3 tpiyrova 1tOU 1ttpttxouv 'tO <TI)µtto A Kat aav to EFG unap

xouv t1tt<TIJ<; 5 · 5 · 5 = 5 3 tpiyrova nou ntpttxouv w <TI)µtio A. :EuvoA.tKa at autft TI)V ntpintro<TI) txouµt 2 · 5 3 tpiyrova nou ntpitxouv w <TI)µtio A. A0poil;ovta<;, txouµt wuA.axtawv

6·53 +2·53 =8·53 =lCXX> tpiyrova ta onoia ntpttxouv w A titt aw tarottptKo wu<; titt navro an<;

nA.tupt<; tou<;.

IlpoPJ...ri µa 3. Na fJp1:80'6v oA.E; Ol TplcIOE; aKEpairov

8pourpa foo pE p11otv Kai o api8po; N paio'U.

( a., b, c ) 3

pE a. >

0 > b > c, 1t01) qo'UV a-

2017 - a. b - b c - c a. dvai TtAf:io TE'Tpayrovo aKE

Ai><JT) :

Aq>ou a + b + c = O , txouµt on 3 a b + b 3 c + c3 a = a3 b + b 3 (-a - b) + (-a - b ) 3 a = -b 4 - 2b 3 a - 3a 2 b 2 - 2a3 b - a4 = = -(a 2 + ab + b 2 ) 2 (1 ) 3 3 3 Enoµtvro<;, av 20 1 7 - a b - b c - c a = k 2 , tott 20 1 7 + (a 2 + ab + b 2 ) 2 = k 2 (k - a 2 - ab - b 2 )(k + a 2 + ab + b 2 ) = 20 1 7 (2)

{k - a22 - ab - b22 =

<=>

{

Aq>ou o 20 1 7 tivm npcino<;, 0a npf.ntt

2 2 2 2 1 <=> k - a - ab - b = 1 <=> a + ab + b = 1008 k = 1009 k + a + ab + b = 20 1 7 2k = 201 8 (3)

fta va tax(}tt a 2 + ab + b 2 = 1008 (4) 1tpE1ttt Ot a, b VCl tivm Kat Ot ouo apnot, OtClq>OptttKU, 'tO aptattpo µtA.o<; tivm 1ttpttt0<;. EmnA.Eov' txouµt Ott 9 1 1008 ' apa 9 1 a 2 + ab + b 2 ' 01tO'tt tUKOACl 1tpOKU1tttt Ott 1tpE1ttt 3 1 a Kat 3 1 b . Enoµtvro<;. ot a, b otmpouvtm µt w 6, onott ypaq>ouµt a = 6 m Km b = 6 n , onott ri (1) yivttm

m 2 + mn + n 2 = 28 (5) Oµoiro<;, Ot m, n npfatt VCl tivm apnot, 01t0tt m = 2x Km = 2y . Tott T) ( 7 ) yivttm x2 + xy + y 2 = 7 (6) fta va EXtt aKtpmt<; A.ucrtt<; TJ ttA.tutaia npfatt TJ OtaKpivoucra ro<; npo<; y va tivm µT) apvrinKi\ Kat tf.A.tto tttpayrovo. 'Oµro<; D.. = x 2 - 4( x 2 - 7) = 28 - 3 x 2 , onott: x2 = 1 it x2 = 4 it x2 = 9 . Enttoit, a 0 ea txouµt x 0 ' 01t0tt x E { 1 , 2 , 3 } . EnttOit y 0 ' naipvouµt 't(l l;tUyl') (x , y ) { (1 , -3), (2 , -3), (3, -2) , (3, - 1) } , onott, aq>ou a = 12x, b = 1 2y txouµt on ( a, b ) E { (1 2 , - 36) , (24 , -36, ) , ( 36, -24) , (3 6, - 1 2) } A6yro tou on a + b + c = 0 Km tou ntptoptcrµou a > O > b > c , txouµt TI'I µovaotKf\ A.u<TIJ ( a, b , c ) = (3 6, - 1 2 , - 24) . n

IlpoPJ...11 µa 4. 'ECf't'ro � 1) 8ETtK1] pi�a Tt); E�icJroa11; x 2 + x - 4 = 0 . To nol..'Urowpo l p x = a.D X 0 + a. n -1 x n - + + a.1 x + a.o ' 01t01) n 8ETtKo; aKtpaio;, EXEl G'UVTEUCf't't; Jlt) apVt)Tl-

( )

•••

Ko-6; aKtpaio'U; K«l api8pt)TlK1] TlJ11\ (i) Na anood�E'TE O'Tl: a.o + a., +

•••

P(;)

+ a.n

2017 .

(mod 2 )

(ii) Na fJpdTE Tt)V EMXlG'Tt) O'UVa'Tl\ TlJ11\ TO'U a8poicJpaTo;: Ai><JT) (i)

Enttoit o apt0µo<;

q = -l + .J0 2

ao + a. + ... + an .

tivm appT)tO<; Kat to noA.urovuµo EYKAEIAHI: B ' 104 T.4/16

( x) = P ( x ) - 20 1 7

EXtt

------ Ma0riµ.a-riKoi Aiaymvurµ.oi - Ma0riµ.a-riKt� 01.llµ.maot� -----, - I - JU ' 01to'tE prp;ou<; <J'\>V'tEAscr'tt<; Kat pi�a 'tOV apteµo ; ' ea EXEt pi�a Kat 'tOV cru�uY'l 'tO'U 2 2 ea (hmpEhat µE 'tO 1tOA'UcOVUµo 'P ( x) = x + x - 4 . Au-ro 1tpOKU1t'tEt aµEcra a1tO TIJV 't(l'U'tOTIJ'ta TIJ<; Sta.ipcmi<; F (x) = P(x) - 20I 7 = (x 2 + x - 4) Q (x) + Kx + A- , a1to TIJV o1toia yta. x = i; A.aµ�a vouµc /(; + A = 0 ' 01tO'tE 1tpOKU1t'tEt O'tt /( = A = 0 , mpou 0 apteµo<; ; Eivm UPPTIW<;. Apa 'U1tUp XEt 1toA.u©vuµo Q( x) -rfaoto ©cr-rE: F(x) = P(x) -20 I 7= (x2 +x-4)Q(x) an x n + an_1Xn-l + ... + a1x + a0 - 2017 = ( x2 + x - 4 )Q(x) (1) A1to 'tTJ crxsmi (1) yta. x = 1 A.aµ�avouµE: a0 + a1 + ... + an - 2017 = -2Q ( 1) a0 + a1 + .. . + an = 2017 - 2Q ( 1) 1 ( mod 2) (ii) 0EcopouµE 'tO crUVOAO { a0 , al ' ... , an } µE cr'tOtXEta µTI UpVTtnKOU<; aKtpatOU<; 1tO'U tK<lV01tOtOUV nl n 'ta 1tapaKa-rco: (a) ani; + an_ 1 i; - + ... + a1 i; + a0 = 2017 (p) 'tO aepotcrµa ao + al + . .. + an dvm 'tO EA.nxtcr'tO Suvmo. Ila.pa.TIJpOUµE 1tpcO'ta on aA.T1BEUEt TI crxsmi : 0 ::::;; ai ::::;; 3 , yta. Knee i = 1 , 2 , - 2. Ilpayµa.n, a.v 11-rav StmpopEnKa yta. Ka1tot0 i = 1 , 2 , - 2 , -ro-rE w crl>voA.o { a0, .. . , ai-1 ' a; - 4, a;+ i + I, a;+ 2 + 1 , a;+3 , ...an } <=>

..., n

ea EtXE cr-rmxda. µTI apvrinKOU<; O.Ktpmou<;, ea. tKUV01totOUcrE TIJ crxsmi (a.), EVOO ea EtXE aepotcrµa crwtxdcov µtKpo-rEpo a1to amo wu cruvoA.ou { a0, a1 , , an } , 1tOU dvm U't01tO. • • •

'Ecr-rco -r©pa Q ( X ) = bn _ 2 x n-2 + bn _1X n-l + ... + b1 x + b0 • TOTE a1tO 'tTIV 'ta'U'tOTIJ't<l an xn + an_1Xn-l + ... + a1X + a0 - 2017 = (x2 + x - 4 )( b11_2 Xn-2 + bn_3 X n-J + ... + b1X + b0 ) 1tpOKU1t'tOUV ot tcrOTIJ'tE<;:

a0 - 2017 = -4b0 = -4b1 + b0 a1 = -4b2 + b1 + b0 a2 a3 = -4b3 + b2 + b1

<=>

= -4bn-2 + bn-3 + bn-4 = bn-2 + bn-3 = bn-2

a0 - 2017 al - bO a2 - bi - bo a3 - b2 - bi

an-2 - bn-3 - bn-4 = -4bn-2 = bn-3 an-I - bn-2 = bn-2

ai+2 - bi+ l - bi = -4bi+2 ' yta. Knee i = 0 , 1 , . .. , - 4 0 ::::;; a; ::::;; 3 , yta. Knee i = 1 , 2, - 2 , a1to TIJV 1tpcOTIJ E�icrcomi a1to n<; 1ta.pa1tavco a0 = 1 Km b0 = 504 . A1to TIJ SEUTEPTI E�icrcomi A.aµ�avouµc a1 = 0 Km b1 = 126 . n

rcvtKU tcrx\>Et on: E1tEtSil dvm 1tpOKU1t'tEt on

= -4bo - -4bI - -4b2 = -4b3

..., n

A1to TIJV -rphTI E�icrcomi A.aµ�avouµE cri>voA.a.

a2 = 2

b2 = 1 57.

LUVEXi�ov-ra<; oµoico<; A.aµ�avouµE -ra

{b0 , bl ' b2 , ...b1 4 } = { 504, 126, 1 57, 70, 56, 3 1, 2 1, 13, 8, 5, 3, 2, l , O, O} { a0, ap a2 , . ..a1 4 } = {I, 0, 2, 3, 3, 2, 3, 0, 2 , I, I, 0, I, 3, I} E1toµtvco<; TI EA.nxtcrTIJ Suva-ril nµil wu aepoicrµaw<; a0 + a1 + . . . + an Eivm 23 . EYKAEIAHI: B ' 104 -r.4/17

-------

Ma011µaTtKoi Atayrovtaµoi - Ma011µaTtK� OA.'ll µnulo� -------

n poKp l f.I OT I KOS li 1 aywv 1 oµos 201 7 8

A11p 1Aiou

201 6

n po P hnw t Aivt:Tat �1rf©vto aKaA1JVO Tpiyrovo ABr qyt)'papptvo m: K6KA.o c(O,R) (JU: AB<Ar l"(IDVO\l JU: ni; n#.a>pti; Br, Ar, AB avria'rotxa. 0 1rEptyE ypapptvoi; K6KAoi; 'TO\l 'Tf>l"(IDVO\l AEZ (tcrrro (c l ) ) 'TEJlVEl 'TOV KUKAo c) crro (Jt)pEio 0 mptyt)'pappt-

( c 2 ) ) TEJlVEt TOV K6KA.o ( c) crro (Jt)JU:io B '. 0 ntptyt)'papptvoi; K6(tcrrro ( C 3 ) ) 'TEJlVEl 'TOV KUKAo ( c) crro (Jt)pEio r. Na MOOEig:TE On:

voi; K6KA.oi; TOO Tptyrovoo BAZ (tcrrro KAoi; 'TO\l 'Tf>l"(IDVO\l rAE

A '.

(

( fJ) To Tt:Tprut#.a>po AEA 'B, dvat E"fYPcl'lflJlO. (jl) Ot EUOE{Ei; M,' EB, Kfll zr cn>VTpqoov.

,'\ lJ(j'l

C1 ) )

I:xt1 µa 1 Ano to eyyeypaµµ£vo (cnov JC6tlo ( n:tpanAf:upo AA'IZ exouµe: AA ' l = AZl = 90° = TA ' I ( * ) . Ano to eyyeypaµµ£vo tEtpanAf:upo r �IE ( ecpocrov rI <>txotoµoi;), exouµe: t

�I = (1) 2 Ano to eyyeypaµµ£vo tEtpanAf:upo BMZ ( ecpocrov BI <>txotoµoi;), exouµe: B �2 = (2) 2 Ano to eyyeypaµµ£vo tEtpanAf:upo B�IB , exouµe: � = B I . 3 Ano to eyyeypaµµ£vo tEtpanAf:upo BB , A , A exouµe: B I = A; = 90° - A� .Apa eivat: � 3 + A� = 90° . Ano to eyyeypaµµ£vo tEtpanAf:upo AEIA , exouµe: A� = A . 2 Apa eivai: �1 + � 2 + � + �� + �� = 1 80°. Auto ITT} µaivet ott to tetpanAf:upo A 'E�B ' eivat eyypa'lftµo. 3 Dµota ano5eucv6ouµe ott Kat ta tetpanAf:upa �ZAT' Kat ZEr'B ' eivai eyypa'lftµa. Apa ot eu0dei; �A , ' EB , Kat zr , cruvtpexouv crtov pu;tKO K£vtpo trov JC6tlrov. A

n po pi.11 µu 2

Na anood;.:n OTt o aptOpoi; p ayoVTa 'TOV

2n+l .

A = ( 4n ) ! , ono'll n !· ( 2n ) !

OtTtKoi; aKtpmoi;, dvm aKtpmoi; Km fxtt na

EYKAEIAID; B ' 104 T.4/18

------- Ma8qpa-r1Koi A1ayrov1apoi - Ma8qpa-r1Kt� 01.llpmao&� (J:J1µGiwm, : O apz8µ6� n ! yza n e N , opi(craz a'lt:o Trf axt<Yf/: n ! = 1 · 2 · ... · n , Kaz 0 ! = 1 .)

--------

Aua11 ( I o.; Tporroc;)

M1topouµe va ypa\j/ouµe tll crxeITTJ A = ( 4 n ) ! = 3n · (3n + 1) (3n + 2) · · · ( 4n - 1) · 4n e Z, acpou 3n e Z . n !· ( 2n ) ! n n �TI') O'UVEXeta 1tapaTI')pOuµe Ott O'TI')V 1tapayoVt01t0l1lITTJ tOU n ! cre "(tVoµevo 1tp0)'t(J)V 1tapayoVt(J)V 0 eK0E-

exp ( n) =

l;J l; J l J

(1) + ... + : 2 o1tou m eivm o µeyaA.Utepoc; q>UcrtKoc; apt0µoc; µe triv t8toTI')ta 2m ::; n . Auto 1tpOKU1ttet a1to TI')V 1tapa Til P11ITTJ on ta 1tOAAa1tA.acrta tou 2 exouv 1tapayovta to 2. Ta 1tOMa1tAacrta tou 4 exouv µia aK0µ11 q>opa 1tapayovra to 2, ta 1tOMa1tAacrta toU 8 EXOUV µia aK0µ11 q>opa 1tapayovta to 2, K. o. K. µEXPt ta 1tOMa1tAaO'ta toU 2m ::; n . A1to Til O'XEITTJ (1) 1tpOKU1ttet on TI')<; tou 2 tcroutm µe

� ( �) ; ;; l J l : j l 2:� J l ; J l ; j l J l�J l��J l��J. l 2��2 J l J l j l J ( ) J exp ( n) ::; +

Me to i8to O"Ke1tnKo ppicrKouµe on: 2 2n exp (2n) = + + ... + 2

+ ... + : = 2 1 =n+

;

1 + + ... +

_1 = n -

(2)

+ ... + : = n + exp (n) 2

( 3)

= 2n + n + ; + ; + ... + : = 3n + exp (n) 2 E1toµEvroc;, 0 eK0ETI')<; tOU 2 O'TI')V 1tapayovt01t0l11ITTJ toU A ea eivm exp (4n) =

3n + exp ( n ) - [exp ( n) + n + exp ( n) = 2n - exp ( n) ;::: 2n - n - : = n + : ;::: n + 1 , 2 2 n+I 01tOte 0 aKepmoc; A exet 1tapayovta to 2 . 2°' Tporroc;

, tOU (2 n ) ' exouµe , A = (2n + 1)(2n + 2) ... (4n - 1 )(4n) . Me TI')V a1t/\.01t0t11ITTJ Ott . , n! fiapaTI')pouµe Ott O''tOV apt0µ11Tft EXOUµ.£ n - 1 apnouc;, a1to 'tOV 2n + 2 Ero<; tOV 4n - 2 , 01tOte pyal;;oVta<; KOtVO 1tapayovta to 2 a1to Ka0e tEtota 1tapev0eITT] exouµe eva.v 1ta.payovta 2 n -l . TeA.oc;, pyal;;ovtac; Km tO 4 0.1t0 'tO 4n , exouµe Ott A = 2 n+ 1 n(n + l )(n + 2) ... (2n - 1) . (2n + 1)(2n + 3) ... (4n - 1 ) n! n(n + l )(n + 2) ... (2n - l ) 2n - l , 01tot A n +I 2n - l · Dµroc; = e =2 (2n + 1)(2n + 3) ... (4n - 1 ) , n! n-1 n-1 01tote o A eivm a.Kepmoc; Km Oimpeitm a.1to t0 2 n+I . '\

npoP"-11 µa 3

( J

f : lR � lR Kat g : 1R � 1R 1t0'll lKaV01tOlOilv T1}V lGOT1}Ta f ( x - 3 f ( y)) = x f ( y ) - y f ( x ) + g ( x ) ' yta KaOt: x , y E lR Kat g ( 1) -8

Bpd·n: oA.&i; 01 J111 Jl.1}0&VlKEi; (fl)Vapn1at:1i;

Aua11 ( I o.; Tporroc;)

• .

E>a. a.1to8eil;ouµe 1tpc0ta Ott 11 cruvapTI')ITTJ f µ1topei va. 1tapet TI')V nµfi 0. Ilpayµa.n, a.v f ( 0 ) = 0 ' auto tcrxt)et. Av eivm f ( 0) = b -:t:. 0 , tote ettovtac; x = 0 crTI') 8e8oµtvri el;icrro<Jll f (x - 3 / (y )) = x f (y ) - y f (x) + g (x) , (1) A.a.µpavouµe: (2) f (-3 / (y)) = -by + g (O) . E1tet8fi w 8el>tepo µeA.oc; TI')<; (2) 1taipvet oA.ec; nc; 1tpa.yµa.nKec; nµec;, faetat on 11 cruvapTI')ITTJ f eivm e1ti EYKAEIAHI: B' 104 -r.4/19

Ma9rip«'TlKOi Aiayrovurpoi - Ma9qpa'TlKti; 0A\lf11tlUOEi;

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tO'U lR , o1tote umipxei c e lR tetoio rocrte f ( c) = 0 . rta y = c O'trl crxemi ( 1) A.aµpavouµe f ( x) = -cf ( x) + g ( x) � g ( x) = ( c + 1) f ( x) , o1tote TI crxemi ( 1 ) yivetm (3) f ( x - 3/ (y)) = xf (y ) + ( c + 1 - y ) f ( x) f ( x - 3 f ( c + 1)) = xf ( c + 1) . ria y = c + 1 O'trl crxemi (3) A.aµpavouµe Av efoouµe I ( c + 1) = a ' t0t€ exouµe I ( x - 3a) = ax' a1tO tTlV 01tOta 1tpoidmt€t TI crxemi f ( x) = a ( x + 3a) , a E lR• Av fitav a = 0 ' t0t€ ea eixaµe I ( x) = 0, yta Kaee x E lR ' ato1t0. A1to tTl crxemi I ( c) = 0 1tpOK1.'>1tt€t Ott a ( c + 3a) = 0 � c = -3a . E7toµevro� exouµe g ( x) = a ( 1 - 3a) ( x + 3a) , a e lR • . A7to tTlV motrlta g (l) = -8 exouµe: a (l - 3a )(1 + 3a) = -8 <=> 9a3 - a - 8 = 0 <=> a = 1 . Apa eivm f (x) = x + 3, g (x) = -2(x + 3) . :Etrl O"Uvexeta euKoA.a €1taA.T1e €Uoµe ott f ( x - 3/ (y)) = x - 3y - 6 = x f (y ) - y f ( x) + g ( x ) . 2°� tporroi;

Av efoouµe o1tou x to y 1taipvouµe f ( x - 3 f ( x)) = g( x) , o1tote ava�Tltouµe o/...e� tt� O"Uvaptficre� rocrte I ( x - 3 I (y)) = xf (y) - yf ( x) + I ( x - 3 I ( x)) o ) KClt €1tt1tAEOV tO'xU€t Ott g(l) = -8 <=> /(1 - 3/(1)) = -8 . (2) Av /(1) = 0 , tote TI (2) oivet ott /(1) = -8 , ato1to. Apa eivm /(1) "# 0 . l <JZlJ fl l<Jµoi; 1 : H avvapr17U1] I dvaz 1 - 1 . A1t00€J4TI · Ilpayµatt, yta x = 1 TI (1) oivet /(1 - 3 I (y )) = f (y ) - yf (1) - 8 . (3) Av f(x1) = f(x2 ) , tote yta (l((>OU eivm /(1 - 3/(x1)) = /(1 - 3/(x2 )) , TI (3) oivet ott f ( l ) ""O

f(x1 ) - xJ(l) - 8 = /(x2 ) - x2 /(1) - 8 <=> xJ(l) = x2 /(1) <=> x1 = x2 , o1tote TI f eivm 1 - 1 . Tropa, yia x = 0 TI ( 1 ) oivet /(-3/(y)) = -yf( O) + /(-3/(0)) . Av /(0) = 0 ' t0t€ 1taipvouµe /(-3/(y)) = 0 = /(0) ' 01t0t€ (l((>OU TI I eivm 1 - 1 , ea exouµe Ott - 3I (y) = 0 <=> I (y) = 0 yta Knee y ' to 01t0l0 eivm at01tO, (l((>OU I (1) 0 . J c;xupt<Jµoi; 2 : Av f (0) 0, 11 avvapr1JU1J f dvaz E:7ri. Ilpayµatt, yta x = O T1 ( 1 ) oive1 /(-3/(y)) = -y/(O) + /(-3/(0)) . (4) ApKei va oei�ouµe Ott av eivm wxov 1tpayµattKO�, U1tapxet Xo ' cOO't€ I (xo ) = Ilpayµatt, fotro eivm wxov 1tpayµattKO�, t0t€ a1tO tTlV (4) PM1touµe Ott apKei va exei Mmi (I)� 1tp0� y TI e�icrromi - yf(O) + /(-3/(0)) = H teA.eutaia oµro� exei Mmi (l((>OU /(0) 0 , €1toµevro� O"U vaptrlmi f eivm e1ti. Acpou tcbpa TI O"Uvaptrlmi f eivm e1ti, u7tapxe1 c rocrte f ( c) = 0 . 0faouµe CJtTlV ( 1 ) 01tou x = c , 7taipvouµe f ( c - 3f (y)) = cf (y) Km acpou TI f eivm €1ti TI e�icrromi f(y) = x exei Mmi (I)� 1tp0� y yia Ka0e x ' 01t0t€ TI t€Ae'Utaia µa� oivet Ott yta Kaee 1tpayµattKO apteµo c2 c , c - x , 7tmpvouµe , , x , 1crxue1 ott I ( x ) = - - -x . , I ( c - 3 x ) = ex , o1tote yia x to 3 3 3 *

AvttKaetcrtcbvta� O"trl (2) 1taipvouµe ott f(l + c - c2 ) = -8 <=>

AvttKaetcrtrovta� exouµe Ott tTlV aPXtKfi .

-3c (1 + c - c2 ) -8 c3 - c + 24 0 (c + 3)(c2 - 3c + 8) = 0 c = -3 f ( x) = x + 3 KClt apa g( x) = -2x - 6 ' Ot 01tOIB� PM1touµe Ott tKaV01tOtOUV =

<=>

EYKAEIAH.2:. B ' 104 -r.4/20

<=>

-------

MaOt)paTlKoi Alayrovlapoi - Ma011paTlKt� 0A1'J.11tUJOt� -------

npoPA.11 µa 4 I:.1'ov xivaKa &ivm ypaµ.µ.tvoi apxiKa Kaxowi 9t1'tKoi «Ktpmoi. Kavot>µt Ka9t q>opa µ.ia axo 1'tc; a KoA.ot>Otc; KtVl]atic; : (a) Av avaµ.taa <JTot>c; apiOµ.oi>c; t>xapxot>v oi>o oiaooxiKoi, t<JTro n, n + 1 , 1'01't µ.xopoi>µt va 1'ot>c;

ap1]aot>µ.t Km va ypcl'f/Ot>µ.t Tov apiOµ.o n - 2 . (p) Av &ivm ypaµ.µ.tvoi oi>o api9µ.oi xot> axtxot>v K«1'a 4, fo1'ro k , k + 4 , µ.xopoi>µ.t va 1'ot>c; ap1]aot>µ.t Km va ypcl'flot>µ.t Tov api9µ.o k - 1 . Ka1'a 1'tl oiapKtia 1'tlc; oiaoiKaaiac;, µ.xopoi>v va xpoKi>mot>v Km apvt)1'tKoi apiOµ.oi <JTov xivaKa. Av OEV µ.xopoi>µ.t va KclVOt>µ.t Kcl1t0l« «XO 1'tc; xapaxavro KlVl]<Jtlc;, " oiaoiKaaia 1'EAElcOVEl. Na xpoaowpiat1'E 1'tl µ.fyi<J1'fl Ot>V«1'1} 1'tµ.1} 1'0t> aKtpaiot> c µ.t 1'TIV aK6A.ot>9'1 10161"11'«: Avt;ap1'fl1'« µ.t 1'0 xoioi api9µ.oi &ivm ypaµµ.tvoi apxiKa, at oA.11 1'TI oiaoiKaaia, 61.oi 01. api9µ.oi xot> dvm ypaµ. µ.tvoi <J1'0V xivaKa va &ivm µtyaA.i>1'tpoi 1} iaoi axo c. A ua11

ea U1tOOeil;ouµe Ott TJ µfytcrni ttµfi tot> c eivat lcrlJ µc -3 . npayµatt, autfi TJ ttµfi e1ttWYXUVetat av l;eKt vficrouµc µe toui; apteµoui; 1, 2, 3, 4, 5 , tote aKoA.ouerovtai; niv otaOtKacria: a

1, 2, 3, 4, 5 � 0, l, 4, 5 � 0, 1, 2 �- 1, 0 �- 3 . (* ) ea a1tooeil;ouµe tropa ott 01tot0t Kat av eivm ot apxtKoi apteµoi Km µe o1tota cretpa Km av Kavouµe tti; Ktvficreti;, Kaveii; apteµoi; ocv dvm µtKpOtepoi; Cl1tO -3 . ea ppouµe µia avaUoironi rocrte Kaee <popa 1tOU e1tavaA.aµpavouµe µia a1tO tti; Ktvficreti; TJ 1tOcronita autii va 7tapaµtvet crtaeepfi. 'Ecrtro w 11 7tpayµattKfi pisa tou 7toA.urowµou P(x) x3 + x 2 - 1 . Tote tax1)et w3 + w2 = 1 � wn+I + wn = wn-2 . Ott (1) 5 E1ticrTJi; to P( x) Otatpei to 1toA.urovuµo Q( x) = x + x - 1 , 01tOte (2). Q( w) = 0 <::::> w5 + w 1 <::::> wk+4 + wk = wk -I A1to tti; ( 1) Km (2) PAi7touµc ott av eeropficrouµe f.vav oc\Jtepo 1tivaKa 1tou avti yta toui; apteµoui; EXet cre Kaee KiVTJcrTJ toui; apteµoui; crav eKe£ni tou w , tote cre eKcivo tov 1tivaKa to aepoicrµa trov crtotxeirov ei vm avaUoiroto avel;apnita µc to av ea e<papµocrouµe to a) cite to p). fta napaoetyµa yta tTJV aKoA.ou eia ( * ) o oc\Jtepoi; nivaKai; ea eiXe toui; apteµoui; P WI w2 w3 w4 w5 � WO WI w4 w5 � WO WI w2 � w-1 WO � w-3 Km yta napaoetyµa, µeta 'tTJV 1tpc0tTJ KlVTJcrlJ to aepotcrµa t(J)V apteµrov 1tapaµtvet to iOio a<pou =

w3 + w2 = wo .

E1toµtvroi;, apKei va l;eKtvficrouµe µe to eA.axtcrto owato aepotcrµa crniv apxiJ (yta to OeUtepo 1tiVaKa), W W 1tOU elVClt µeya/\.Utepo a1to, wI + w2 + w3 + w4 + . . = = = w-4 . '

'I '

1 - w w5

E7toµtvroi; crtov MA.o 1tivaKa Kaee apteµoi; eivm µcyaA.utepoi; a1to -4, 011A.aofi 2::: -3 . A47.

To 1t0AUIDVUµo P( x ) = x3 + px + q EXel tpcti; Ota,<popettKEi; µctal;u toui; 1tpayµattKEi; pisei;. Na a1tooeil;ete ott p < 0. N 4 1 . 'Ecrtro d ( n ) o apteµoi; trov eettKrov Otatpetrov tou eettKou aK£pmou n . Na a1toOeil;ete ott:

(! .!. !)

(

)

+ + ... + :::; d ( l) + d ( 2 ) + ... + d ( n ) :::; n l + ! + .!. + ... + ! . 2 3 n 2 3 n A20 . 'Ecrtro A a\JvoA.o µe n crtotxeia. Bpeite to µcyaA.Utepo owato apieµo t>1toauvoA.rov tou A 1tou eivm tfaota rocrte va µ11v u1tapxet Ka1toto a1to auta 1tou eivm t>7toa\JvoA.o Ka1totou aA.A.ou. n

EYKAEIAHI: B ' 104 T.4/21

HOMO MA THEMA TICUS H Homo Mathematicus c:ivm µHl oriJA.11 oto 1teptoOtK6 µ(l(;, µc: oKo1t6 TI']V avtaUaY'l a1t6\jleCOV Km tt]V avamu�11 1tpo�A.11µa·noµou 1tUV(l) Ota e�ii<; 0tµata: 1 ) Tt eivm ta Ma0f1µattKa, 2) Ilpfaet ii oxt va OtOUOKOV'tCll, 3) Ilotot c:ivat ot tlaoot tcov Ma011µattKffiv Kat 1toto to avttKeiµevo tou Ka0ev6c;, 4) Ilote<; c:ivm ot c:<papµoytc; tou c;, 5) Ilote<; c:m oriJµec; ii tlaoot c:mott]µffiv a1tattouv KaA.ii yvc0011 to>V Ma011µattKc0v yta va µ1topfoc:t Ka1toto<; va touc; o1touoaoc:t. crovraKTtKlj e1mp01clj : Kspauapii517c; I'zavv17c;, Maviar:olCOVAov AµaAia, M�).,zoc; I'zwpyoc;, M7Cpov(oc; ErtAzoc; I. n dvw

rn Ma.011µ0.nKO.;

To napaKa'tco Keiµevo eivm 0µ1A.ia nou €Kavs o Kcovcr'tav'tivo<; Kapa0so5copi]<; <JT11V EA. ATIVtKi] Ma0T1µanKi] E'tmpsia cr'tt<; 1 9/5/1 924, µs 0€µa TI'IV 51C>acrKaA.ia 'tCOV Ma0T)µa't1Krov <JTI'IV 8su'tspo pa0µm sKnai8sucrri . H 8aKruA.oypacpTl<JTI €y1vs an6 TI'IV Ilsp1081Ki] 'EK8o<JT) Ma0T1µm1Krov :Enoue>rov «E>saiTI'l'to<;». (OK'tropp10<; 1 987, Hpatls10 Kpi]T11 <;, EK86T11 <;-81su0uv'ti]<;: Mavd>A.T1<; MapayKUKT)<;). To Ksiµevo 8aKruA.oypacpiJ0T1KE an6 sµa<; (JS µoVO'tOVlK6, 8tCl'tTIPIDV'tCl<; TI'IV 1tpCO't6'tU1tT) op0oypacpia Km 'tO yA.cocrcrtK6 i8icoµa T11 <; snom<;.

7CpoJ.cyoµcvo.

Kwv(fro.vrivov Ko.po.01::0<5wp 1j «llcpi rw v Mo.OtfµO.TIK<bv cv 'TI/ Mimi EK7Co.1&v(j£1»

«E>' apxicrco TI'IV oµiAiav µou 'taUTilV µs 'to spd>T11 µa: «81Cl'tt 51McrKOV'tm 'ta Ma0TlµCl'ttKa st<; 'tCl crxoA.sia µa<;;» A€ycov 'ta «Ma0T1µm1Ka» 8ev evvoro m crw1xsia tTI<; ap10µT1'ttKi]<;, m onoia sivm pspaico<; anapaiTI'l'ta s1<; Ka0s av0pconov avi]Kovm Et<; ocrov8i]no'ts oA.iyov nsnoA.mcrµ€vov nsp1pat..A.ov, aUa TI'IV fscoµs'tpiav Km TI'IV AvaA.umv. EmcrKonrov n<; TI'IV Icrwpiav 'tcov noA.mcrµrov cruvav'ta A.aou<; tKava npoT1yµ€vou<;, oh1vs<;, co<; 01 A'ts€Km n.x. , s1 Km 61..co<; aµotpOl µa0TlµanKIDV '}'VID<JeCOV, e8T1µt0Up'}'T1crav Kpmma<; noA.t'teia<; Km crri µavnKi]v 'tEXVT)V. OU'tCO<; CVl<JxUe'tm TI t8fo 6n 'tCl Ma0TlµanKa 8ev eivm ocrov cpav'tas6µs0a anapaiTI'lm 8ta 'tTIV avanru�lV µm<; 1te1t0Al<J'tt<JµEVT)<; KOlVCOVia<;. Ilp0Ks1µ€vou A.01n6v nspi 'tTI<; ev 'tTI µ€crri EKna18sucrs1 818acrKaA.ia<; 0a T18uvaµs0a va cpavmcr0roµev ronov crxoA.eiou a1tT)Uayµ€vou Tll <; 818acrKaA.ia<; 'tCOV Ma0Tlµa'ttKrov, µa0i]µmo<; avnKa0tcr'taµevou 81 aUou avacpsp6µevou s1<; aA.A.ov tla8o Tll <; av0promvri<; yvrocrsco<;, co<; n.x. st<; TI'IV B10A.oyiav, 0scopouµEVT)v ev Tl1 KUpia au'ti]<; crri µacria co<; Emcr-riJµTIV Tll <; scoi]<;. Kata TI'IV sµi]v nsnoi0T1crriv o K1lp10<; Myo<; 8m tov onoiov m Ma0T1µanKa, oµou µsta Tll <; 818acrKaA.ia<; wu op0ou xs1p1crµou Tll <; µT1tp1Ki]<; J'AID<J<JT)<;, anotsA.ouv to K€vtpov papou<; Tll <; ev Tl1 µ€crri EKnm8sucrs1 818acrKaA.ia<; st<; 61..ou<; wu<; crUJ'XPOvou<; A.aou<; Tll <; EupronTI<; Km Tll <; Aµsp1Ki]<;, sivm anA.oucrtam TI napa8ocrt<; Km TI cruvi]0sm. �16n, co<; ano8s1ms1 TI nsipa, tinots ev Tl1 81anMcrs1 tTI<; av0promvTI<; Ko1vcovia<; 8ev sivm co<; EK Tll <; cpucrsco<; autou <JUVT11 PTl't1Krotspov wu crxoA.siou. EXPstacr0T1 n.x. 0MtlT1pO<; o tlovmµ6<; wv onoiov sn€cpspev s1<; TI'IV \j/UXtKiJv 81a0smv Km TI'IV vootponiav tcov

av0proncov µia raU1Ki] Enavacrmcrt<;, 6nco<; 5uvT10iJ va crmµancr0i] vfo<; rono<; crxoA.eiou co<; TI 1tPIDTl1 «Ecole Centrale» ppa8U'tspov µswvoµacr0eicra «Ecole Polytechnique» i] «Ecole des Ponts et Km 6crm aA.A.m t8pu0Tlcrav KCl'ta TI'IV snoxi]v sKsivriv un6 TI'IV sni8pacr1v 'tOU Monge, µaA. A.ov npocop1crµ€vm 8m TI'IV napacrKsui]v av0proncov 'tCOV scpT1pµocrµ€vcov smcrTI'lµrov. Ta npoypaµµma 61..cov 'tcov anavmxou avaMyou cpucrsco<; crxoA.sicov cp€pouv fat cri]µspov TI'IV crcppayi8a 'tCOV eK1tm8sun Krov wu'tcov t8puµa'tcov. Enicrri <; 5uvaµs0a, xcopi<; icrco<; va 1)1tUP�TI Kiv8uvo<; 81mvwcrsco<; U1t6 µ€Uo v'to<; tcr'top1Kou, va icrx;upm0roµev 6n 'ta Eupconai'.Ka crxoA.sia Tll <; µE<JT)<; EKnm8cicrsco<; avayov'tm acp' ev6<; 81a 'tcov crxoA.sicov Tll <; Avayevvi]crsco<; ev l'taA.i a, anva unfoTI'lcrav TI'IV sni8pacr1v 'tcov sK Kcovcr'ta vnvoun6A.sco<; EUT]vcov 8t8a<JKaA.cov au'trov, acp' s 't€pou 8s 8m 'tcov crxoA.sicov 'tou Busav'tiou s1<; 'ta crxoA.sia Tll <; AA.s�av8p1vi]<; snoxt1<;, 'tcov onoicov 8u vmm va 0scopT10d>cr1v an6yovo1 · sivm 8s cpucrtK6v 6n cl<; 'tCl teUAsutaia tauta 'ta Ma0T1µanKa 0a Ka teixov npcotsuoucrav 0fo1v, acpou TI rscoµstpia napa wt<; 'EUT1crtv s0scopsho, Km 81Kaico<;, co<; TI µ6VT) t6tE tEAslCl smcr-riJµTI · aAA.co<; te Km 'tO 6voµa au't6 wuw µaprupei 6n m Ma0TlµanKa anstsA.ouv TI'IV K1lp1av µa0T1crtV Km co<; sK 'touwu K1lp1ov µa0Tlµa. TsKµi]p1ov T11 <;, co<; s1pi]tm, µsyaA.TI<; <JUVT11 PT1ttK6Tll to<; tcov crxoA.sicov sivm Km to sni mrova<; aµsta PA.Tltov tcov npoypaµµatcov Tll <; 818aKtfo<; UAT1<;, napaT11 pouµevov navwu, i8ia 8s ev AyyA.ia, 6nou TI fscoµstpia 81aM<JKstm an' su0sia<; an6 w Keiµs vov wu Eutlsi8ou.

Chausses»,

EYKAEIAH:E B ' 104 T.4/22

f?W.1. tO l , 0: :IHVI:IV)(,\:il

-odu AC!JAD)ll AC!J1lt9nrf i\001 �)ltct Soo30<)."( Sltl mg umiA.13gndnu Jlt3 'Slp..nn S(,tXotd3lt Smrf l}. SnJd13rfoo3J Sltl noJD"(Dd>3)l S9A3 ADJ"(D)lDng1g Altl mg Alp.. lt9mi Ao1 l)IAXno t3J1)lrf1013 nA Slp.. W..Lt0n)l o oo]Jrf oA 01nAc;i.glt ne ·

·Aod31 -9001d3lt m)l i\Od31<}XD1 13J1)ldct0)l i\01 l)IW 'A<)lClD J310ltOAD)ll t\3g 'c;i.01lt9nrf no1 AOd?d>mgt\3 01 J3At)l t\3g SnJd0030 Sl)ldlt? Sltl StOC!JM Lt m)l S1oroA9rf ltArf -oun H ·Sndg?A."(V Sltl m)l Sn1d13rfoo3J Sltl nrflt91)lrf 01 S13 AC!J)ltmrf Lt0nw i\001 ADJ"!D)long1g Altl 1)11D)l c;i.01 -lt9nrf no1 S10�01)1 Lt So1ltl1ndnun mAJ3 u9 C!JAOd<J> "DJd0030 1)ldlt? U m)l 10lt -<).1 10 i\(\OA.JAltDlD)l St)ll)ITIOlt SC!JXCllong ADJOltO Altl 'A0011)lrfA.ndu Aoo1 n1ono Lt t3A.cpd>mg Di\ 310lt?gno 310 -C? Soo1c;i.o m1�01)lg1g Di\ U"!<). Lt A01ltlJndnun mAJ3 ·ltl1)11ono"lltD mAJ3 S1?13g9un Lt ClOJOltO ClOl '( O<).g A9rf 91dn i\01 1)111))1 AOA?rflt?nlt ClOl AC!Jrf)ln Aoo1 So0lJ."tu 01 Sodu m1c;i.001 c;i.01nn AC!Jdg3 Aoo1 So0lJ."tu 01 1)11D)l AoA?rf lt?nlt nodg?ct"!Olt 31ou

-l}.gno10 AC!Jd>ndo)l Aoo1 So0lJ."!lt 01) i�1n3 no1 nrfltd -(!)30 01 i\D}d13TI03d310 i\Ul St3 m13JJA01 Di\ Slt!>Jlt3 ·Aoo1lp.. -ooou AooA3rf 9]monodnu 310101)1)13 AOOl St3D1)110mg m m1AC!Jd0030 Di\ 'DJdm)lCl3 m13J1)110ClOdDlt Sl)ll)IOO '3101/\l)llt i\D}"tll)lOl)gtg Dl13lt?13rf Altl St3 m)l 'i\0030 -7)11omg i\001 Jlt3 c;i.01lt9nrf no1 l}.Xooodu Lt m1n10Jd>3 DA Sood?1tmg1 Soo"l9 nrf ltloc;i.o A<))l1d13rf 01 m13)1!)1)lg -tg ClOlt<) Cl0J3"l0Xo <).0)lli\UTI3 no1 <).OlctD ,ltD 13lt?dlt 31011\l)llt ·x ·v . AC!J1U0Drf i\001 l)ldDlt i\0011)lrfA.ndlt i\001 Soo31h.l}."l\1AD Snd?1ndn3 Sooltl)l)l ADJA.dnotrf ltg A<)lDt\(\g 01 1)11D)l Alll St3 DlAOTil)lgrf no m)l ndlp..)lndnX DAU A<))llt\3A. Dli\Od?d> Dl<)i\OA.3A. S13 ClOl SD}"(D)lOl)gtg Sltl Sod?rf A9)lultdoo39 01 1)11D)l Sood?11mg1 Su UA?rfm3 Di\ A3rfo]JrfoA Aorftlt<))lO ADJ"l AC!Jforfdnd>3 Aoo1 S91)l3

·norfD<))l oo?? no1 nA3rf9Amd> n1 t\(\Olt?tg torf -9A }O)lUDtilt9nrl u9 'Aod3rflj.D i\0101X1)l"!Cl01 'A3rlod> -?d1 ADJOltO Altl Soo3101u Sltl Jlt3 S91ADlt odu m13JJO -ng 'S3AOrllp..ollt3 10 i\Od3rll}.D A3TiOJJ1DrlltXo i\D}OltO Altl 'notiO<))l <).0)llt\3rl13)lllt\l) ClOl i\(!))113 Lt u9 A3TiOOlt -?"lgndnu DA t3Jt?du t\3g u91v ·Slj.oo] Slj.)lu)lndu Sltl m)l norlo9)l <).O)l1d3100?3 no1 nt\3rl9Amd> n1 Sodu S130 -(!JM no1 S1)l)lUDrf lt9nrl Sm UJJ13Xono Di\ S9AD)ll m1 -n1010n)l 'ADJonrf ltD Altl lt]Jrl0mo nA 3101/\l)llt Al0?0 S13 mA}3 t\3g AOO}OltO i\001 'S13DC!JM S1)l)lUUdoo39 SC?"tlt -n 1)11)lOltD , A JlAD 'Slp.. lt9nrl o Soou9 AOO?ti AOA<Jrf 01 mAJ3 SDJ"lll)l!>l)g1g Sltl Stoi\ne <).311))1 ltlc;i.mo1 mw ·m1J3"l31 ltlc;i.n OOJOltO 001 t\3 AOTI1)lgtd3lt 01 Sodu Aoot\3rf 9]otidnoodu A<JlDt\(\g 01 1)11D)l AC!Jforf dnd>3 , lg Alj.oo] Alj.)lu)lndu Altl Sodu lp..nn m1lt?gAno DA �,t -l)IAD C!JAOdd> mm101dnu 'DJ"(D)lODgtg Lt Sod9d>oudn)l AOTil)lrf ltAJngoun Soou9 'tg1)1TI3 t\3 SooA?rf1d)l3)lfo:s: ·1ndod>mg }D)llltOl m ltlh.9un A<).09d>lt"! DA A<)lDt\(\g mAJ3 t\3g n1d13rfoo3J Lt Soo Alj.d>dorl Altl noJA.nu 0091 Somrfl}.9nrl DJ"!D)l!)Dg1g ltl lp..nn t\3 1?TID 'nJd>ndA. -003J Lt AOJO Somrfl}.9nrl 1Xno nnn)long1g lt1 t\3 310(!) 'SnJXdnug: Sltnl)I Sltl l}. Sl)llrf Sltl i\<))ll"l(\ i\<))lUU0Drl 01 9un DJ3rf ltD no91 St3 A0030<)ltCllt\3 m)l i\00301)110 -ndnu Soo3Jh.9un 9un 13d?d>mg 'mAlJ.0V m mJO 'Soo -3"l<Jlt<tO"!DA.3rl Smrl A<))lt"ln A<))lUlt9nrl 01 u9 ntiA.13g -l)ldnu AoA?rf1d)l3)lA.no A3 A3rf oog1)1"! DA mg ·x ·u S?AD -odu mi\}'3 ·Sn}"(D)l!)Dg1g Alj.d>dorl i\OltCll<)3d310 Alp. -nn Altl <).Oli\Dlt 13X?dDlt A3Ti0"!?0U 'i\0011)lrltindfodlt i\001 i\O}Dlt\3 01 i\0"!9 , dnlt AD 'S?gD"tgllt3 Oll}. D0 m)l JDl<tD m <).Oli\Dlt mi\}3 t\3g 'DJ"!ll)lODg1g U m1J3"l31 Sn , d>n 'm)ll}.eAno m 'Al!A?rln131)l3 <)."!Olt tX9 'Srna Lt ·X·u Soo 'ltti9)ln ADdC!JX St3 m)l So1AOTI1)lg1d3u no1

A1ondgJlt3 Altl SC!J)lUon)l,{,nAn m1n101d>n Slp..nn l}.,torl -dnd>3 ltl A3 DJ"!D)l!)Dg1g Lt '119 Su m1l}.rf net\3 DA mA -}3 �A.l)IAD S101c;i.01 Sodu ·c;i.01lt9nrl no1 A1oud1)11D)l ADd?1<t"!D)l A<)lDt$\g 01 1)11D)l 01 Sodu SD}"(D)lODgtg Sltl So1nti<).3Alt no1 Jd3u Lt Slt"!<). Sn?1)lng1g Soow<).3g -mlt)l3 SltD?rf Sltl StOJ3"tOXo Sto1 A3 Sltl A0011)lrlrlnd,t -odu i\001 Jd3lt l}.9©3)10 Di\ <tOA?Til3)l0dlt 'i\<)TI"(D0 -d>o Odlt St1 ltX? Di\ AOTil"t?<f>OO ADJlti<)t\3 D0 SDJOltO Sm 'm?gl m SJnrlrlnd,t SJD)llt\3A. t\3 mAJ3 m1c;i.moi ·Ao -Jg A<))lU)lDdlt Ao1 mg AC!JAU AOOJgo<h3 Al,tXodnu Altl S13 ntil)I AOOlt?"tgoun 'S9)luood>dorl Soo}d($l mAJ3 Soo1 -c;i.o S9uo)lo o AOJlADAno .L ·nd?1<t''(DA.3rl 31oul}.gooouo mAJ3 nJA.o"tnAn Lt nou9 'm3)l<).V l)l)lU)lndu n1 Su llD?dm?3 AD '1)ld)l1ti ADJ"l mAJ3 'Aoo1 S130C!JM l)l)lUDti -Lt0nw m S13 Sn1 i\Od31<).gndg i\(\00l}.10ltOtiloltdX n0 S3AUJO 'i\001\}3)13 i\(!)1U0Drl i\001 DJfo"!DAD Lt ltti9)lD l}.1 -nn m)l . AC!J)lUntilt9nrl Aoo1A<JTI3rf lj.ro)londnu Lt mA -}3 A3g Soo30<).3gmu SlOJ3"tOXo S101 A3 Slt1 S9uo)l!) O ·Sltlc;i.n1 Sltrllp..o -mg: Sltl Soo3?<).lltDAD nod9d>oudn)l Sltl 'AC!J1lt9nrl i\001 Snpn"llt Sltl nodC!JD Sooltl)l)l no1 m)l noA9dX Soo -31h.J3TI3 Sltl )13 Soo 'AOlDA<).gn 01 mg c;i.orlo1A.oy <).O)l -uood13uy no1 Sl}.A.00A.no13 Slj.)lunrlltlono AOOJ3)lnv AC!J)lU)lndu Aoo1 S13?1?1 Sn1n1�31 Sn1 St3 A101)11 Altl C!Jd0030 i\Ot\(\gA})lllt3 AOTil)lf'\l . Al}.OOJ Alt9lj.i\no Altl S13 SC!JA1d3tilt9n)l Su 1)11/\DltD ADJouo Altl 'Soowlp.. -dnAno Sltl moAA? Lt ·X·u mAJ3 ltl<;imoi ·m1uo10A10)l i\}3lt13 A<)g3Xo i\Od3tilj.D i\DOltlAlp..D)l Dtilp..)l i\<)i\10)1 <).Od)lDti <)ltD mOl)llDDll))l S3i\U}D 'AC!JlOi\t\3 i\OOd?lOO -31\ i\D}"(D)l!)Dg1g Altl 1)11D)l l}.A.00A.no13 Lt 01}3lt0)1!)3ltl) 'Slj.1fo mAJ3 13d?rl A3 Sul}. 'Sltl<).Dl Soo3olj.i\Dt Sltl mv

{5oXg3J. 0(1.3rlgol.ltodJ/. OUJ 3rlYJ.O<J.3l.OOrllt9 aoJ/. vrl.OVW<JJ/.V OJ. J3() aOyO)IV1

·1o)lunrllt9nrl lOtiltD1)11g m)l An?1dlp..03un SDJOltO Sn1 'JD)luo1d31003A S1301)11 'DJ"l"!DJ m)l DJAnrfd3J A3 n1g1 'Anolt9l}.dltl3dnu ADJ13noo)l13 ADJD1�31 Altl 1)11D)I ------ SI13IJ.V:W3H.LV:W O:WOH

------

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HOMO MATHEMATICUS

PA.11µfrrrov crxenKcOV 7tPOs to ev Myro Ke<paA.cx.tov KCll ev avarKfl µe 'tflV ofooucra Poiieetav ii 'tflV cruvepya criav wu Kae11r11tou. AA.A.a oev mcrteU(J) on Mvatm va yivetm OtOaoKa A.ia trov Mae11µanKc0v rocpeA.tµos Km anoteA.eoµa nKiJ aveu 'tfls Moeros acpeovrov aoKiJcrerov Km Ka taUiiA.rov npoPA.11µatrov uno trov µae11tc0v. Eav oe o XPOVos ota tllv oioaoKaA.iav trov Mae11µattKrov eeropeitm OXt £7tapKfJs iva acrKcOVtCll Ot µae11"tai e7tapKcOs, ea iit0 (JJ(07ttµotepov Va eA.attroeei 11 7tp0s OtoacrKaA.iav 7tpooptsoµBvri uA.11 iva OtOaOKCOVtm oA.tyotepa µev eeropiiµma Km µtKpotepas eKtacre ros eecopia, aA.A.a apKetai acrKfJcrets. 'Exro a7tOKOµi cret 'tflV evronrocrtv ott ev EA.A.not napaµeA.eitm 11 acrKf1ots trov µae11trov ets w va A.oyap1asouv ano µviJµ11s KCll va Mvavrm va l':KteA.root anA.as apte µ11nKas npa�ets ano µviJµ11s · Km touw eivm voµi sro ev Ol>OlcOOes eMttroµa 'tfls OioacrKaA.ias. Ilpfaet va e�aCJKfleii o µa0fl"tfJs ano trov ta�ecov wu EA.A.11vtKou crxoA.eiou, 07t0l> OtOUCJKl':tCll 'tllV Ap1eµ11nKiJv, iva AO"(aptas11 ano µviJµ11S Km OXt Va ypa<pll anA.as acpmpfoets ii noUanA.cx.macrµous eni napaoeiyµan. Enetoii avecpepa 'tflV Apteµ11nKiJv, eupicrKro TilV eu Katpiav va cr� yvcopicrro on, cos enapa"tfJp11cra, to np6ypaµµa tcov Ma0flµanKrov eivm pepapuµevov noM µe w µae11µa 'tfls eerop11nidts Apteµ11"ttKTJs · Bepairos 11 Ap1eµ11nKiJ eive a<petfl pia trov µa911µa nKrov emcrtf1µrov · ota 'tfls eerop11ttKfJs Apteµ11"ttKTJs, ev euputepa crri µaoia nA.eov, ave1troxe11 Km 11 eero pia trov CJUVap"tfJcrecov, eivm oe 11 Ap1eµ11tt1CTJ 11 11yens trov µae11µanKrov tlaorov. A)..).JJ. wuw noM otacpepet ano w Kata nooov npe nei w µae11µa 'tfls eerop11nKiJs Apteµ11nidts va c5t McrKf1tm ets ta ruµvama. �ton nav o,n evvoei o µae11µanKOs oev eivm l':UKOAOV oute Ol>VCl'tOV va evvoiicret 0 µae11"tfJs 'tOl> ruµvaoiou, oev eivm oe Km OKomµos 11 OtOaotlia tOl> µa0iiµat0s tOO'O eu

pecos, acpou eK tu)V µae11trov 'tOl> fuµvaoiou eA.ax1crt0t ioros ea CJ1t0l>Oacrouv Ma0flµanKa, aUa KCll autoi ea µopcpro0ouv ets t0 Ilavemcr"tfJµtov etOtKcOs. �ta 'tOUtO eiµm 'tfls yvc0µ11s Ott 7tpe7tet va 7t£ptopt oeii 11 uA.11, iitt� otoa(JJ(etat ano 'tflV eecop11nKiJv Apteµ11nKiJv e� ta A.iya anapm"tfJtros avayKaia µ8P11 au"tfJs, va ev1crxueii oe ot>tro 11 OtoaoKaA.ia trov µae11µanKcOV Til s reroµetpias Km 'tfls AA.yeppas acpou Km 11 UAll autrov 7t£ptopicre11 CJUµµetpros, ro ote va Mrovtat tKavai acrKiJcrets uno trov µae11trov. TeA.os eupicrKro A.iav pepapuµevov to 1tp6ypaµµa trov Mae11µattKcOV Ota ta AuKeta. Km auta oev npoKettm va napaoKeuaoouv µae11µanKous, voµi sco. AA.A.a 1Cl>piros npoKettm eKei va cpoi"tfJcrouv oi µeAAOVtl':s Va £1tlOoerootV et� £1ta'Y"fBAµata 11 e� e mo"tfJµas CJUvoeoµevas otevros µe ms eettKas em cr"tfJµas. �ta 'tOU'tO 11 uA.11 tCOV Mae11µanKcOV Ota ta crxoA.eia auta voµisro on 7tpi7tet va nepixet 7tOAAas ecpapµoras ainves eupicrKovtm ey"(Utepov µe tov piov 'tOl> avepro1tOl> Km 7tp0� tO µeAAOV £7tayyeA.µa 'tOl> µae11wu trov oxoA.eicov O.l>tcOV. Outro n.x., evro ea toxupiset0 KOVeis on ets 'ta AUKeta ea OtOaCJKfl tm 11 eerop11nKiJ Apteµ11nida euputepov, ey© voµi sro on £Ket aKpt�cOs ea 110UV't0 Va Ael'lf11 evteA.ffis 11 otoaoKaA.ia wu µaeiiµmos wuwu ro� eA.axmta crKomµos. Tauta 7tpe7tet VO excoµev l>7t0'1f11 11µrov, roote av o V'tffis ta npoypaµµma eivm empapuµeva ets pae µov roote v' a7tOKAet11tCll 11 <ruµ<pcOVIDs 7tPOs 'tO ill s avco nveuµa OtOa(JJ(aA.ia, npoteivco oncos 11 Etmpeia µas KataPaA.11 naoav napa t:Ots apµooiots 7tp0CJ7ta eetav 1tPOs anA.07toi11oiv autoov. Ev K11cptoia, Maios

1924»

{7C11Y �: w Keiµevo aVTO µac; w 1Capaxwprwav, ave�O.pTYJTa 0 ivac; a1C6 WV illo, Ol m]µavw.:oi uvva&A<pOl EaKYJc; A11COpt5t(YJc; (7tp6et5poc; wv "Movueiov KapaBwt5wp�") Kl o IlavayidJTYJc; Xpov61t'ov).oc; (yvwcrr6Tepoc; we; "Ilapµevi'5YJc;" 1j, Ka< 'illovc;, we; "NdJe TYJc; EvKki&1ac; frwµeTpia<;"))

II.

"EvKJ.d&ia TuuµeTpia, ayaTC'l µov "

Dnros 1tpoavaneiA.aµe (CJ'tO 7tp0fl'YOUµevo ), ano 'tO teUXOs 'tOU'tO, ea OtaKO'lfOUµe 7tpooropt va 'tfl 011µocrteuCJf1 yeroµetptKcOV evvotcOV Km avt' aut0u ea 011µooi£Uouµe cre tpe1ts CJUVBXEIBs, onouoaia eupiiµma ano 'tllV npcotoru1t11 epyacria, wu crri µavnKou CJUvaMA.cpou h,. Ap"(Up11 Kavteµip11, nou exet titA.o «Ilros µewtpenetat 11 Emnec5oµetpia cre :Etepeoµe� tpia» (Aeiiva ). :E11µetrovouµe on Otatflpouµe 'tfl crUVta�ll Km 'tO "(AroCJotKO t8iroµa a •Xl'ill• t r 'tO\) 7tprotO'tl>7tOl>, oµros "(ta tEXVtKOUs Myous oev Otatf1pouµe 'tflV opeoypacpia 'tOl>. A' 1CpoJ.ey6µeva

1971

0.1CO TOV 1CpoJ.oyo WV avyypa<pia : «l:K07t0s 'tfls 7tapOUCJfls epyacrias eivm 11 napouoiams Km 11 unooe�ts tp07t0l> anA.ou Km cracpous, Ota 'tOl> 07t0lol> nacra 7tpotaots 'tfls Emnec5ou reroµetpias OUVCl'tCll va anetKovtoeei Km otaronroeei cos avticrwixos

7tp6taots tlls fecoµetpias WU Xropou. EK wuwu Kaeicrtatm cra<pes, on 0Mtl11pos 11 E7ti7tEOOs reroµe tpia µeta trov �troµatrov. trov eerop11µatrov Km trov npoPA.11µa-

EYKAEIAHI:. B ' 104 -r.4/24

crx1'i11a

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HOMO MATHEMATICUS

--------

'tCOV nii; ouvcmu va anEtKovtcred Eti; wv fEcoµE'tpt- 'tpia wu Xropou» Kov Xropov 'tcov 'tptrov 8tacr'tacrEcov coi; µia f ecoµE0.7CO rn; ,, Em:!;11 y1j aw; f(at l:vµpoJ,1apo1) c; ", ro v avyypa<pia :

epyco 'tOU'tCO EKneEµtvcov, 8i8oµev mi; KU'troei E1tE�T]yi]crE1i;: Dnou ava<pepoµEv "Eueuypaµµa 'tµTjµCl'ta" eEro pouµcv on mum ava ouo opisouv opeoyrovtov. o) Ta npicrµam eEropouv'tm roi; opea npicrµam, 'trov onoicov m aKµai ava ouo opisouv opeoyrovia Kai ot K'6A.tv8poi eEropouv'tat coi; "opeoi K'6A.tv8poi", 'trov 01tOlCOV Ot U�OVEi; ava OUO opisouv opeoyrovta. E) At e8pat 'tCOV 1tptcrµa'tCOV ovoµasovmt µE 'ta ypaµµam 'tCOV aKpcov µtai; 8taycoviou nii; e8pai;. To µfapov '!OU Eµpa8ou nii; e8pai; cruµpoA.isE'tat µE E Km Ka'tro 8E�ia 'tou E ypacpoµcv m ypaµµCl'ta µn; 8tayroviou nii; e8pai;.

«Ilpoi; nAT]pEcr'tepav KCl'tUVOT]O'lV 'tCOV ev '!CO

aT) �iE8pot yrovim wu npioµawi; eEropouvmt m OiEOpot 'tCOV napanl...EU pcov EOprov auwu . » TeA.oi;, o cruyypacpfoi; µai; Ef;T1YEi ncoi;, yta va naµe ano 'tTJ 81cr81acrmni Euilii8Eta fEcoµE'tpia npoi; 'tTlV tptcrOtUO''tCl'tll , apKd va avnKCl'tClO''!TjO'OUµE 'tTl Ae�TJ "crri µdo" µ& 'tTl Ae�TJ "rueEia", 'tTl Ae�TJ "EUeEi a" µE 'tTl Ae�T] 11E1tl1tEOo", 't11 Ae�T] 11E1tt1tEOO" µE 'tTl Ae�T] 11 0''tEpEo", 'tTl Ae�T] "µi] Koi;" µE 'tTl Ae�T] "Eµpa86", 'tTl Ae�T] "EµpaM " µE 'tTl Ae�T] "oyKoi;" , 'tTl Ae�T] "nl...Eupa" µE 'tTl M�TJ "e8pa" .

7Capa&iypara

lo. c5urc5uiO'TaTI/ EV1clei&1a I'ewµerpia: «'tO a epotcrµa 't(J)V 'tE'tpaycOVCOV 't(J)V Kaee't©V 1tAEUpcOV opeoycoviou 'tptyrovou tcrou-rm npoi; w 'tE'tpayrovov 'tTli; U1tO'tEtVOUO'Tli; aU'tOU» (cr:x. l ) rp1ac5ui.O'rar'1 EV1cleic5e1a I'ewµerpia: «W a.epot crµa 'trov 'tE'tpayrovrov 'trov Kaefarov EOprov op0oyro viou npicrµmoi; tcrou'tat npoi; w 'tE'tpayrovov 'tTli; U1tO'tEtvOUO'T]i; e8pai; UU'tOU» (ax. 2) 2o. c510'c51aO"Tar'1 EvKkic5e1a I'ewµerpia: «'to 'tE 'tpayrovov nl...Eupai; 'tptyrovou, TJ onoia KEi'tm aneva vn O�Eiai; yroviai;, lO'OU'tat npoi; 'tO aepotcrµa 'tffiV 'tE'tpayrovrov 'tffiV OUO aUrov nl...f:uprov '!OU T]Aa't'tCO µevov KCl'ta w 81nA.amov ytvoµcvov 'tTli; µiai; EK 'trov nl...Eu prov au'trov Em 'tTlV npopoA.i]v 'tTli; UUT]i; nl...Eu pai; E1ti 'tClU'tTlV»( O".X· 3)

rp10'c5uforaT11 EvKleic5e1a I'ew µerpia: «'tO 'tE'tpayroVOV µtai; e

111. A 11 n) ro !;i:p a n::

Ilocra OEK<lOtKa \jlT]q>ia '!OU 1t XPT]crtµonotd T] NASA;

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Opai; opeou 'tpt')'©VtKou npioµa wi;, TJ onoia KEi'tm anevavtt o �dai; 8tt8pou yroviai; auwu, t crou'tat npoi; 'tO aepotcrµa 't(J)V 'tE 'tpaycOV(J)V 't(J)V 860 UAA(J)V Ebprov ClU'tOU, T]Aanroµtvov KCl't6. 'tO 8nA.amov y1voµEVov 'tTli; µ1a;i; EK 'trov EOprov au'trov Eni 'tTlV npopo A.i]v 'tTli; aUT]i; eopai; E1t' auTftv» (cr:x.

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OT]µocrrnuouµE to OEU'tEpo µtpoi; wu w µepoi; Ei:xaµ& OT]µocrrnuoEt crw npomouµevo cv8iacpepovwi; crri µEtroµawi; nou µai; fo'tEtAE o cpi- 'twxoi;). Iltcr'twouµe nroi; dvm :xpi]mµo va �avaod A.oi; 'tTli; cr'ti]A.T]i; Ilavaytffi'tT]i; OtKovoµaKoi; (w npro- 'tE m op1crµ1Ka crtot:XEia wu nprowu µepoui;. 7CpoJ.<701wva

c5evrepo µipor;: )..tjµµa 1 '

Ilavco cr'tti; aKµei; ox,oy,oz, 'tptcropeoyrovia� 'tpiE8pT]i; yroviai; o.xyz, naipvouµE 'ta crri µEia A,B,f av'ticr'tot:Xa. To'tE tcrx\>Et (ABf)2= (0Bf)2+(0Af)2+(0AB)2 (cr:x. 5) aTC0&1�'1 Ai; dvm M TJ opei] npopoA.i] wu 0 crw EninE8o (II) '!WV A,B,f. To'tE TJ AM npoEK'tEtvoµf:VTJ 'teµvE1 Ka9Em TIJV Bf crw M 1 (acpou TJ Bf Eivm opeoyrov1a crni; EUeEiEi; OA,OM wu Emne8ou OAM). ME Pacrri 'tO eE©pT]µa 't(J)V 'tptrov Ka9farov ea exouµE OM 1 .l Bf, apa T] � 01= � AM 1 0 ea dvm T] ClV'ttO''totXT) Ent1tEOT] 'tll i; oi EOpT]i; 'trov Emne8cov (A,B,f),(O,B,f). Ano w opeoyrovto, crw M, 'tpiyrovo M1MO e:xouµe EYKAEIAHI:. B ' 104 'T.4/25

-------- (--J (( ))

HOMO MATHEMATICUS

2 2 MM 1 MM 1 MM 1 2 auve1= i) auv e1= = 2 ( ) OM 1 OM 1 OM 1 Ano to opeoyrovio (crto 0) tpiyrovo M1MO, exouµE (OM1)2=(MM1).(AM1) naipvouµE 2 ( MM 1 ) (MM, ) 2 m>v e,= ( MM 1 }( AM 1 ) ( AM1 )

(2)

Ano (1),(2)

( )

Ano (3),(4),(5),auvexouµE 2 e1+ auv2 e2+ auv2 e3= _

EnEtoi) m ei. e1, e3 Eivm m avticrLOtXE<; EninEoEc; yroviEc; tOOV oiEoprov nou O'XT)µatisEl to EntnEOO (A,B,r) µE ta EninEoa (y,o,z),(x,o,z),(x,o,y) avti crtmxa Km ta tpiyrova OBr,oAr,OAB, Eivm npo poA.Ec; tou ABr crta EninEoa (y,o,z),(x,o,z),(x,o,y) tl'J<; oocrµEVT)c; tpIBOpT)<;, ea EXOUµE:

(3)

(MM1 ) + ( MM 2 ) + ( MM 3 ) ( AM 1 ) ( BM 2 ) ( rM3 )

( )

------

(OBr)=(ABr)auve1. Dµota (OAr)=(ABr)auve2 Km (OAB)=(ABr)auve3 Tic; tEA.EutaiEc; uwrovouµE crLO tEtpayrovo Km nc; npocintOUO'E<; tO'Otl'JtE<; npocreetouµE Kata µEAT) Kl EXOUµE: (OBr)2+(0Ar)2+(0AB)2 ( ) =(ABr)2(auv2e1 + auv2e2+auv2e3)

)

7 , µE(0Br) pacrri triv (6), oivEt triv anooEtKtfa 2+ (0Ar)2+(0AB)2 =(ABr)2

V. "A 1Ca vr11a11 aw «A vro w �ipare;»

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tlJ

ea npOKU'lfEl ea EXEl a7tOKAl0'11 mto tT)V nµi) nou ea npoeKUntE av µnopoucmµE va noUanA.amacrouµE µE tl'JV aKptPT) ttµi) tOU n ea T)tav EKatocrta! ME A.iya Myta to µi)Koc; tl'J<; nEptq>EpEta<; EVO<; citlou µE DtaµEtpo tplO'EKaLOµµupta XtAlOµEtpa ea i) taV Meoc; Kata EKatocrta, 6cro to µi)Koc; LOU µi Kpou oaKWAOU crac;.

40 4

[1t11ril :

https://liveyourmaths.com/2 0 1 7/02/04/%cfl>/o 80%cfl>/o 8c %cfl>/o 83%ce%b 1 %ce%b4%ce%b5%ce%ba%ce%b I %ce o/ob4o/oce%b9%ceo/oba%ceo/oac %cfl>/o 88%ce%b7%cfl>/o 86%ce%afl>/oce%b 1 %cfl>/o 84%ce%bfU/ocfU/o85-%cfl>/o 80%cfl>/o 87%cfl>/o8 l %ce%b7%cfl>/o 83%ce%b9%ce%bc%ce %bfl>/ocfl>lo 80%ce%bf%ce%b9%ce%b5-2/]

EYKAEIAHl: B ' 104 't'.4/26

��........ .. ..,,;,.,.,__ _ .

Ta�fi :

Ym;'60l>VOt

A' AYKEIOY

Ta;11�: Xp.

Aa<;:a12i811c;1 Xp. Tcrt<pUKH<;

Ka!crouA.11c;

E11avaAn11r1 Kt<; AaKf\0£ 1 <; AAyE�pa<;

AO'KflO'fl l : a) Av x + y + z = va mroodl;tTt oTt :

Kat

xyz * 0 TOTt

1 1 1 + 2 2 2 + 2 2 2 =0. 2 2 2 x +y -z y +z x z +x -y Av x · y · z = l , Ka\ l + x + xy * O TOTt va _

p) mroodl;tTt OT\:

x + y + z =1. 1 + x + xy 1 + y + yz 1 + z + zx 1 1 1 y) Av x · y · z * 0 Kat - + - + - = 0 , TOTt x y z y+ z z+ x x+ y . a1tOOttl;tTt OT\ : -- + = -3 . + z y x ----

: x + y + z = 0 => x + y = -z => (x + y) 2 = (-z) 2 => z 2 = (x + y) 2 => x 2 + y2 - z 2 = x 2 + y2 - (x + y ) 2 = -2xy :t: O Oµoiroi; y2 + z2 -x2 = -2yz :t: O, z2 + x2 -y2 =-2zx :t: O . Me avnKmacrracrri crro npffiro µEl..oi; TIJi; sTJtoUµf:Vll i; crxscrri i; sxouµe: 1 1 1 + 2 2 2+ 2 2 2= 2 2 2 x +y -z y +z -x z +x -y 1 1 1 1 x+y+z -+ -- + -- = · · · = - - · -2xy -2yz -2zx 2 xyz = - .!_ · _Q__ = 0 . 2 xyz 1 p) 'Exouµe : xyz = 1 => xy :t:. 0 Kat z = - . Apa xy 1 1 x xy l l + y + yz = l + y + y= l + y +- + + :;t O Km xy x x 1 xy l x l + z + zx = l + - + x = l + - + - = + + :;t O xy xy xy y xy Me avnKmacrmcrri crto npffiro µEl..oi; TIJi; s11touµf:V11 i; crxscrri i; sxouµe x y z = = + + 1 + x + xy 1 + y + yz 1 + z + zx x 1 x xy = + + =1. + yx + 1 + x + xy 1 + x + xy 1 + x + xy 1 + x + xy , : -1 + -1 + -1 = o => yz + xz + xy = o y) Icrxuei xyz x y z Aua11 : a) Icrxl)e1

----

---

Kovoµ11� ApTt

=> yz + xz + xy = 0 =>xz+xy= -zy=>x( z + y) = -zy => z+y = - � . (1) . Oµoiroi; an6 TIJV iota crxscrri x x z + x zx x + y xy , f3ptcrKouµe : -- = --2 (2) , -= --2 (3) . z z y y Me np6cr0ecrri Kata µEA.11 t(J)V crxfoerov (1), (2) Km y+z z+x x+y z:y zx xy , (3) exouµe: -- + -- + -- = --2 - -2 - x y z x y z2 3(z:y)(zx)(xy) (z:y) 3 + (zx) 3 + (xy) 3 = = 3 (xyz) (xyz) 2 3(xyz)2 = 3 . (Icrxl)et TJ np6tacrri : Av a+ f3+y = O, (xyz)2 tote a 3 + f33 + y 3 = 3af3y ). *

AO'KflO'fl 2: AivtTat f1 1tapacnmr11 :

2 A (x -6x+2 9)(x3 -81x) x - 3x

a) Na pptiTt yia 1t0lt� T\J1E� TOl> x opi�tTat 11 1tapacna0'1] A . p) Na a1tl..o1tot1)ant TflV 1tapaaTaO'fl A y) Na /..:6atTt TflV tl;iaro0'1] A = o) Na \)1t0Ao"(latTt TflV 1tapacna0'1] A av TO

x = (-1)201 7 - (-1)2016 .

Aua11 : a) ria va

opisetm TJ napacrtacrri A npsnei 2 Km apKei: x - 3x * 0 . 'Exouµe: x 2 - 3x = 0 <=> . . . <=> x = 0 it x = 3 . Apa TJ napacrmcrri A opisetm yia Ka0e x e = IR - {0, 3} Km µ6vo · .. p) fta tTJV napacrtacr11 A iaxl)ei : 2 - 6x + 9)(x 3 - 8 1x) (x - 3) 2 x(x 2 - 8 1) A - (x x 2 - 3x x 2 - 3x = (x - 3)(x 2 - 92 ) = (x - 3)(x - 9)(x + 9) . y) fao sxouµe: A = O � (x - 3Xx -9)(x + 9) = 0 � x E {3, 9, -9} � x E {9, -9} . H nµit x= 3

-- �-'-----'--

Tipocrocrxfl cri; µta EKq>pacrwa'J AE1ttoµepi;ta. fta µta 'tO D, O'tClV Aiµi; O'tt opi�Etm ym Ka0i; x e A , EVVoouµi; 6tt A � , OllMIOi] ocv Eivm Kat' avayKllv A=D. Av 6µro<; 1touµi; 6r1 opi�i;rm ym Ka0i; x e A Km povo yt' auta, EVVoouµi; 6n A=D. •

(Jl)VUPTilcrrt µi; 01'.>vo/...c) optcrµou

EYKAEIAHI: B'

104 T.4/27

Ma811Jux'T1Ka y1a 'T'IV A' At>KEio\>

anoppimetm (oev eivm Mcrri ) mpou 3 ie D . &) YnoA.oyisco npoota TilV nµfi tou x. 'Exouµe: = -1 - 1 = -2 .Me x = ( -1 )2011 ( -1 )2016 avnKatacrtacrri tou x <Jt11V napcicrtacrri A txouµe : A = (-2 - 3)(-2 - 9)(-2 + 9) = ... = 385 . AaK11 a11 3 : AivovTm 01 t;1aroat� :

3x 2 + 2(A - 3)x - 2 = 0 (1) Km -2x 2 + 7x + µ - 1 = 0 (2) µE A,µ e 1R .H t;iaroa11 (1) fxt1 avTi8tTti; p�ti; Km 11 t;iaro<Jll (2) txt1

aV'Tia'TpO<pti; pisti;. p) Av

a) Na ppd'TE 'T� T1µti; Trov 7tapaµiTprov A. Km µ.

A. = 3

µ = -1 Kat X p Xz dvm 01 pi�ti; x 2 + (A + µ )x + µ - A = 0 'TOTE :

Kat

'T'li; t;iaro<Jll i; i) Na ppd'Tt 'T'IV 'T1J11\ 'T'l i; 7tapaaTa<Jll i;

A = (3x1 - 2)(3x 2 - 2)

ii) Na l..t>at'TE 'T'IV t;iaro<Jll :

4 4 X 1 + -) X2 I x - 3 + += 0 . (3) (x - 3)2 + (I Xz X 1 X1 Xz Aua11

)2 - 2X1X 2 (x = ... = -3 ' � + _i_ =4 (x1 +X2 ) = 1 + X2 X1X2 X1X2 X1 X 2 = ... = 2 . Me avttKatcicrtacrri tcov napancivco napacrtcicrecov <Jt11V e;icrcocrri (3) txouµe: (3) � (x - 3) 2 - 3 lx - 3l + 2 = 0 Ix - 31 2 - 3 Ix - 31 + 2 Ix - 3 1 = co Ix - 3 1 = co � co 2 - 3co + 2 = 0 co = 1 Ti co = 2 � lx - 31 = 1 Ti l x - 3 1 = 2 � x e {4, 2} fJ X E {5, 1} � X E {l, 2, 4, 5}

�

0 � }� } =

AaK11a11 4 : Aivnm 11 t;iaro<Jll

x 2 - 2(A. - l)x + A 2 - 3 = 0 , A. e JR (1) .

a) Na ppd'TE y1a 7towi; T1µti; 'T'li; 7tapaµi'Tpol> I..

11 t;iaro<Jll (1) EXEl ot>o pi�ti; 7tpayµa'T1Kii; K«l av1ati;. p) Av A < 2 K«l X 1 ,x2 01 OtlO pi�Ei;j 'T'IG; t;foro<Jll i;j (1) 'TOTE : i) Na EK<ppaat'TE roi; (J\)VUP'T'l<J'I To\> I.. 'T� 1tapaa'Taat�: x1 + x2 , x1 x , x + x i , x i + xi . 2 ii) Na ppd'TE y1a 7towi; T1µti; 'TO\> I.. 1axt>t1 :

Ene18fi 11 e;icrcocrri (1) txe1 avti0ew; pise� : icrx()et : 8 � 0 Km S O :Exouµe : � = [2(A. - 3)]2 - 4 · 3 · (-2) = ... = xi + x i = x1 + x 2 • (5) 2 4(A. - 3) + 24 > 0 nou icrx()et yta Kci0e A E lR . iii) Na 7tpoao1opiaE'TE 'Tli;j 'Tiµti; 'TO\) A roan va 1<JXVE1 : x1 (x1 + 2x2 ) + x 2 (x 2 - 2) > 2x1 • (6) -2(A. - 3) => A. = 3 . S = 0 => -� = => =0 Au cn1 : a) fta va txe1 11 e;icrcocrri (1) Mo pise� a 3 Enicrri � 11 e;icrrocrri (2) txei avticrtpoq>e� pise� npayµanKt� Km civicre� npfae_t Km apKd 8 > 0 ' Exouµe : � > O � [-2(A. - 1)] 2 - 4 · 1 · (A. 2 - 3) > 0 icrx()et : 8 � 0 Kat P 1 :Exouµe : 41 � 4(A. - 1) 2 - 4/... 2 + 1 2 > 0 . · � = 7 2 - 4(-2)(µ - 1) � 0 => . . . => µ � - 8 p) fta A. < 2 mo epooTilµa a) ano8ei;aµe 6n 11 µ-1 P = 1 => -- = 1 => ... => µ = - 1 nou enaA.110ei'.>et e;icrcocrri (1) txe1 Mo p\Se� x , x 2 npayµanKt� Km -2 civtcr�. i) 'Exouµe: x 1 + x 2 = -� = ... = 2(A. - I) ( 1 ) , 41 a µ � - - . n.pa µ = - 1 . ' Kat 'tTlV O.Vt<JCOITT) 8 x 1 x 2 = l_ = A. 2 - 3 (2) , x ; + x; = (x 1 + x 2 )2 - 2x 1 x 2 Jfopan't p11a11 : Av 11 avicrcocrri � � eivm McrKoA.o a va em.Au0ei, tote anA.oo� e;etcisouµe av enaA.112 2 2 0ei'.>etm ym TilV nµT) µ=- 1 nou ppT)Kaµe an6 TilV = [-2(A. - 1)] - 2(A. - 3) = ... = 2A. - 8A. + 1 0 (3) , x: + x; = (x 1 + x 2 )3 - 3x 1 x 2 (x 1 + x 2 ) P= l p) i) Me avttKatcicrtacrri tcov nµoov A. = 3 Km = [2(A. - 1)] 3 - 3(A. 2 - 3)[2(A. - 1)] µ = -1 11 e;icrcocrri yivetm x 2 + 2x - 4 = 0 Km = ... = 2(A. - l)(A. 2 - 8A. + 1 3) (4) x1 +Xz =-2 , x1 · Xz = -4 . 'Exouµe: A= (3x1 -2X3x2 -2) ii) Me avnKatcicrtacrri tcov napacrtcicrecov (1 ) Km = 9(x 1 x 2 ) - 6(x 1 + x 2 ) + 4 = = -20 ( 4) crTilv (5) txouµe: ( 5) �2(A.-1XA.2 -�+13)=2(A.-1) X2 X1 + .. y 1tO/\.oy1�ouµe 'Y n) ' npcota ' n� napacrtacret� � (A. - l)(A.2 - 8A. + 1 2) = 0 E {1,2, 6} � X 2 X1 A = l , ene18fi A < 2 . x; x; + 4 4 � � = iii) EKteA.oovta� n� npci;e1� crto npooto µtA.o� Til� = Km - + - . 'Exouµe: + (6) Km avnKa01crtoovta� n� ( 1 ),(3) naipvouµe: ( 6 ) � x; + x; - 2(x1 + x 2 ) + 2x 1 x 2 a)

•

. . .

EYKAEIAHI: B '

104 T.4/28

� . . �A >0

MaOriµaTtKO: yta TflV A ' AvKtiov

<::> 2A. 2 - 8A. + 1 0 - 2[2(A. - 1)] + 2(A- 2 - 3) > 0 <::> ... <::> A. 2 - 3A. + 2 > 0 <::> (A. - l) (A. - 2 ) > 0 �< 2 <::> A- - 1 < 0 <::> A. < l . A.aKq<nt

AivETat fl <f1>VUf>Tfl<r11 f µE TiJJto:

f(x) =

�.

x - x-6

a) Na pptiTE TO 1tEOio optaµo'6 T1)i; <f1>VUf>T1)<J11 i; f. P) Na ppdTE Ta miµda Toµl]i; T1)i; ypmptKl]i;

1tap6:GTa<J11 i; T1)i; <f1>VUf>T1)<J11 i; f µE Tovi; O:l;ovti;. y) Na ppdTE yta Jtowi; Ttµii; T01> x 11 ypmptKl] ' 1tap6:oTa<J11 T1)i; f dvm KO:Tro a1t6 TOV O:l;ova x x. 0) Av X 1 = f(l) Kat X 2 = f(4) , Va ppdTE

tl;iaromi i-0u pa9µo'6 1t01> £xtt roi; p�ti; Tovi; apt9µo'6i; x1 Kat x2 e) Na vJtol..oyiaETE TflV Ttµl] TTti; 1tapaaTaarii; : •

A = f (2) + f (2) l + f(2) 1 - f(2)

•

fta va opisemt 11 mwaptrt<JTJ f npenet Km apKei : 5 - j x j � 0 Km x 2 - x - 6 * 0 . 'Exouµe: 5 - Ix! � o <=> . . . <=> -5 x x 2 - x - 6 = 0 <::> ... <::> x = -2 it x = 3 A.pa w m::8io optcrµou trt� eivm w cri>voA.o A = [- 5, -2 ) u (-2, 3 ) u (3, 5 ] . p) Ot ttµ11µEve� t(l)V cr11µeirov wµit� µe toV a�ova x'x enaA.110euouv trtV e�icrrocrri : f(x) = 0 . 'Exouµe Aua11 : a)

� �5

•

�

= 0 <=> �5 - lxl = 0 x -x-6 <=> x = -5 it x = 5 . A.pa ta crri µeia toµit� trt� ypaqnKT)� napacrmmi� trt� cruvaptrt<JTJ� f µe tov a�ova x'x eivm ta M 1 (-5, 0) , M 2 (5, 0) . H temyµEvfl to'U crri µeiou toµit� µe toV a�ova y'y

A.otnov f(x) = 0 <=>

eivm : y = f(0) = - J5 . �11A,a8it to crri µeio toµit� 6 trt� ypacptKT)� napacrmmi� trt� cruvaptrt<JTJ� trt� f µe tov a�ova y'y eivm to N(O, - J5 ) . 6 y ) H ypa<ptKT) napacrtacrri tll� f eivm Katro ano tov a�ova x 'x av Kat µovo tcrxf>et f ( x ) < 0 . l:to A exouµe: f(x) < O

<=>

�

<0 x -x-6 lxl < 5 <=> �5 - Ix! > 0 <=> <=> x2 - x - 6 < 0 x2 - x - 6 < 0 -5 < x < 5 <=> <=> -2 < x < 3 -2 < x < 3 o) YnoA.oyisouµe nprom tou� api0µou� x1 , x 2 •

}

'Exouµe .. X 1 - f(l) _- ••• _- - -1 Kat X 2 - f(4) 3 = ... = .!_ .To a0potcrµa s Kat to ytvoµevo p eivm : 6 1 1 S = x 1 + x 2 = = - - , P = x1x 2 = ... = -- . H 6 18 e�lcrOOITTJ 2-ou pa0µou 1tO'U EXet pise� to'U� apt0µou� x1 , x 2 eivm : x 2 - Sx + P = O <::> x 2 + .!... x _ __!._ = 0 6 18 2 <:::> 1 8x + 3x - 1 = 0 . e) YnoA.oyisouµe nprom trtv ttµit f(2 ) . 'Exouµe :

�

= . . . = - fj .Tote µe avttKatacrtacrri 2 -2-6 4 TTt� f (2) <>trtV napacrmmi A exouµe : - J3 - J3 J3 f(2) f(2) A = = 4 + 4 = ... = _8 . + l + f(2) 1 -f(2) 13 J3 1 + J3 1-4 4 f(2) =

A.aK11 a11 6 : Ot otaaTO:atti; x, y tvoi; op9oyroviov flETaPcilloVT(ll ' ETGl cOGTE fl 1tEpiµnpoi; T01> Va 1tapaµivtt aTa9tpl] Km dvm imi µE 24 (aE m). a) Na EK<pf>UGETE TO y roi; <f1>Vclf>TflG1l T01) x Kat GTfl G1>VEXEl(l va ppdTE TOV TiJ1tO E = f(x) 1t01) oivEt TO tµpaoov E T01> op9oyroviov roi; <f1>VUf>Tfl<J11 T01> x. p) Na aJtood/;ETE oTt To tµpaoov E Jtaipvtt Tfl µiytGTll Tlflll yta x = 6 K(ll va ppdTE Tfl µiytGTfl Tlflll T01>.

Aua11 : a) Av x, y eivm 0t 8mcrtacret� tou op0oyroviou tote 11 nepiµetpo� 11 Km to eµpaoov E 8ivovtm ano tt� crxfoet�: I1 = 2x + 2y (1) . E = xy (2 ) 'Exouµe: I1 = 24 <=> 2x + 2y = 24 <::> y = 12-x (3 ) . Me avttKatacrmmi tou y crtrtv crxecrri (2 ) naipvouµe: E=xy=x(12-x)= ... =-x2 +12x . Enet8it x, y eivm 8mcrtacret� tou op0oyroviou x O x O <=> 0 < x < 1 2 . A.pa icrxf>et : > <:::> > y>O 12 - x > O o wno� tou eµpa8ou tou op0oyroviou ro� cruvaptrt<JTJ tou x eivm E(x) = -x 2 + 12x µe cri>voA.o optcrµou to (0, 12) . p) H cruvaptll<JTJ E(x) = -x 2 + 12x eivm napapoA.it µe a = -1 Km naipvet trt µeyicrtrt ttµit yta

{

x = -P = ... = 6 . na x 6 ano trt crxecr11 (3) 2a ppicnwuµe y = 6 ' 811A.a8it to op0oyrovto exei to µeytcrto eµpaoov omv yivet tetpayrovo . H µeyicrtrt nµit wu eµpa8ou eivm: E(6) = -6 2 + 1 2 · 6 = ... = 36 (cre m2) .

EYKAEIAHI:. B'

104 T.4/29

MaOqµanKtl y1a TT)V A' Al>Ktio'U

E11avaAn11TI KE�

AaKtiatl�

rtWIJETp ia� Mixa>..11 � NaKo�

f\aKriari t ri . AivtTat TtTpaxJ..w p o ABrA Kat t. aTco M,N Ta µt.aa TCOV oiaycovicov Ar, BA «VTl <JToixco�. Av AA=4 Kat Br=6, va od�ETE OTl TO µ'l]KO� T01) TJ.l'l]Jl«TO� MN=x J11tOpd va xapel µ6vo 4 aKf.paw� Ttµf.�. A uari : 'Ecmo l) E>etro

Mf{Br . K to µfoo tOU AB. TOte MK = 3, NK = 2 (EUeuypaµµo tµftµa nou EVcOVEt ta µfoa Mo nl..Eupffiv tptyffivou) Kat an6 tptyrovtKft avtcr6tT1ta crt0 tpiyrovo MNK : 1 < x < 5 . Ot aKepmEi; tiµei; wu x crw (1, 5) Eivm ot 2 , 3, 4 .

Axo TO A q>tpvco Ka8tT11 xpo� TflV AB, xo1> Tt µvtt TflV Br aTo A. Av BA=2Ar, va ppeiTt Tfl

ycovia r = x A uari : Av M A

•

t0 µfoo 'tTli; B�, t6tE ea tcrx(>Et AM=BM=M�=Ar KCll a<pou MBA = MAB = 20° ea BXOUµE, ( roi; E�COtEptKft yrovia CJ'tO tpiyrovo ABM). Ano t0 tcrocrKE"Aei; tpiyrovo AMr naipvouµe x = AMr = 40° . �

..-...

�

2Q°40°x

ACJKflCJfl 4 '1• l:TO clKpO r Tfl� oiaycovio1> Ar, op8oycovio1> ABrA, q>tpco Ka8tTfl, 11 oxoia Ttµvti Tt� xpotKTaatt� Tcov AB,AA <JTa <JflJlEia l:,P a-

,,...

VTl<JToixco�. Ad�TE OTl: rBP = I:M" . ""

http://mathematica.gr/forum/viewtopic.php?f=40&t=52292&st art=660#p274952

'Ecrtro A!!,. I I Bf' . T6tE to ABf'!!,. Eivm tpane�to Km t0 Eivm t0 tµftµa nou EVcOVet ta µfoa 6 - 4 = 1 . TE11.tKa, � ' tOU. i\pa ' t(l)V utayrovtrov x = -ot 2 meavei; aKepam; nµBi; tou x i::ivm oi: 1, 2, 3, 4 . 2)

AaKttari

ABr

2fl .

(A. = 90° )

AivtTat µt

op8oyrov10

AB < Ar

Eav r B P = 0 Kat L � r = co , t6tE acpou rB / /AP Kat r� / /AL , exouµE � P B = 0 Km � r B = co . ApKEi /..om6v va &Ei�ouµE 6n � P B = � L B ' 011/..a&ft 6n 'tO tEtpan/..EUpo �PrB Eivm EYYPU'l'tµo.

A uari : /\

Tpiyrovo

Kai txi Tcov xJ..w

prov Ar, Br xaipvo1>J1E avTtaToixco� <Jflpda A,E, TtTota cO<JTE �E = M = AB . Av, E TO µfoov Tfl� Br, va ppEiTE Tfl ycovia

,...

r=x Aum1 : Ta tpiyrova AEf, ME Eivm tcrOCJKEATt µE npocrKEiµEVEi; yroviEi; 'tTli; �acrri i; toui; icri::i; µe x . 'ExouµE E� = 2x ( roi; E�rotEptKft yrovia crt0 tpiyrovo A�E). Av M to µfoo 'tTli; Ar. �

T6tE t0 tpiyrovo �ME i::ivm opeoyffivto ( ME II AB ) AB �E µe ME = = . 'Etcrt, 2x = 30° ==} x = 1 5° . 2 2

,,...

f\aKriari 3fl. AivtTat Tpiycovo ABr µt B = 20° .

Ilpayµan: P L B = Ar B ( o�EiEi; yroviEi; µi:: nl..Eupei; Ka0EtEi;) Kat A r B = � A r = A � B ' acpou Ar= B� ==} OA = OB = or = o�. Apa p L B = A � B 01t0tE �PrB eyypa'l'tµo. f\aKtt<n1 sl] - l:To tacoTtpiKo u1oaKtJ..oi>� TP •rro/\

V01) ABf (AB = Ar ) JlE

EYKAEIAHI: B ' 104 -r.4/30

.....

96° ' 1>1tclPXEl <JflJlelO

PBr 1s· Km pfa

roaTt

yrovia BAP -.

Ma&qpanKa r•a 'TllV A' At>Kf:iot>

30· . Na ppti1't Tl1

http://mathematica.gr/forum/viewtopic .php?f=20&t=57257 A

H ycovia BAr µnopi::i va 0i::cop110i::i i::niKi::-

Aua11

vtp11 crwv K6KA.o (A, R) µi:: R = AB = Ar .

A<JKll<Jll 7ri, AiV£1'(ll 1'piyrovo ABr Jl.f:

K«t

Ar =

OlX01'0Jl.O;

1'0t>

AA.

A -.

AB + BA ' 1'01'£ va ppd1'£ Tl1 yrovia

Aua11 : Ilpoi::Kti::ivco niv AB Kma BE = BLl -ro-ri:: AE = AB + BE AB + BLl = Ar . Ano TIJV tcrO'tT)'ta

f Ll = 3 =1 0 � = 40"

-rcov tptyrovcov Mf, ME (IT - - IT) txouµc Af AM B &: = 0 ' 01tO'tc AB 20 Km ano . w Tpiycovo ABr txouµc 60° + 0 8 ° 0 A

P Ll (156° ,12° ,12°) , BPLl =

Av o K6KA.oc; (A, R) -rµilcri::t TIJV f crw , tote cr:mµa-risov-rm m -rpiycova: To tcrocrKi::Aic; Ar� 'tO tcronAf:upo MB , 'tO tcrocrKi::Aic; (84° ,48° ,48° ) , onoti:: w -rpiycovo BAP i::ivm tcrocrKi::Aic; nic; µopq>ilc; 78° , 78° ) , cruvi::m:Oc; x 78° .

(24°,

f\aK11a11 6'1• Aivt1'«t tva looxAf:t>po Tpiyrovo ABr. Mia Et>Oda (t), 1} oxoia ottpxt1'«t axo 1'0 r Of:V £:xt1 KOlVcl Gt)µ.da Jl.f: TllV xl..f:t> p a AB. 'E (J1'(l) M 1'0 µ.iao T11 ; AB. Axo 1'� KOpt><pt; A K«t B <ptpvro Ka0t1't; xpo; TllV (t), 01 oxoit; T11V Tt µ.vovv G1'a A K«l E av1'iG1'01xa. Ati;1't 01'1 1'0 1'pi yrovo AME dv«t laoxl..f:t> p o. http://mathematica.gr/forum/viewtopic.php?f=20&t=5 72 1 8

fM .l

Eivm AB , µta nou w M i::ivm w µfoo nic; AB . Ano -ra i::yypli'l'tµa -ri::-rpanAf:upa BEIM(BH' +rMB=l80") Km MM'A(AM"'=Mf'=90°) npoK61ttct on Km MEr = MBf = 60° ELlM MAr 60° , 0111..aoil w -rpiycovo LlME civai tcronAf:upo. A uari :

.......

......

.....

60°

http://mathematica.gr/forum/viewtopic.php?f=20&t=57 l 3 2

Ll = =

.....

,.. r

l<JXVtl

f\aK11a11 sri. !:To £YY£Ypaµ.µ.tvo 1't1'paxl..f:t> p o

ABrA 1'0t> a:xfaµ.aTO;, Ta M, N dV«t 1'a µ.taa 1'(l)V AB, rA aV1'1G1'oixro;. Aivt1'«t 01'1 rBA ""

NMP . .-...

30°,

rA AP =

•

ppd1't Tl1 yrovia

.....

http://mathematica.gr/forum/viewtopic.php?f=20&t=565 5 l

Av 0 TO Kmpo Tou K6KA.ou, TOTE OM .l AB apa KCll 'tO <l or i::ivm tcr01tMropo. Apa Km onon: w -ri::-rpanAf:upo 0a i::ivm i::yypli'l'tµo A 6011 :

fOLl = 2fBLl = 60° ON .l fLl , Ll OLlP=120° �LlOP=OPLl=30° NOP=60°, OMPN /\

EYKAEIAHE B' 104 T.4/31

Ma9flµaT1Ka y1a TTIV A' At1Kdot1

NMP """

B L1 r = Ar K = ro (o�Ei.Ei;; µE Ka0Em;; nAf:up&<;). AKoµJJ: ro + cp = ro + e = 45 ° => cp = s => Kr = Ar

yuui O M P + ONP = 90° + 90° = 1 80° . Apa = NOP = 60° . ,..._

B , •porro<;

<l>epvro to U'f'O<; /\

tou •ptyffivou AB..1 .

'Ex.ouµE: Z A K = AK H = ro co<; EVT6<; EVaAAa�. E1tt<IT)<; ZAE = EAr = ro an6 'tJJ yvrocrTJ1 np6m<ITJ U'f'OU<;-OtX,OTOµou-otaµETpou 1tEptKUKAOU, 1tOU ayo V't<lt an6 'TJJV ilita Kopu<pi) (Eoffi 'TJJV A). A

,,/

----

/,.,.--�-------

AaKJJ<Hl l OJJ.

Aivt:'Tat opOoyrovto ABrA. 'E<nco K TO 0'1]µdo "

T0µ1\� 'T'll � OlXOTOJ101> 'T'll � ycovia� A Kat 'T'll � mro To r KaO&To'V <n11v BA. At:i;Tt: 6T1 BA=rK. A

Apa to Tpiyrovo rAK EtVat tO'OO'KEAE<; Kat rK = Ar = B..1 A6yro TT)<; tcr6'tl)Ta<; Trov &tayrovirov tou op0oyroviou. r •porro<;

\ \

<l>epvro an6 to K 1tapaAA11A11 O''tl) rL1 1t0U TEµVEt m; Br,M crm E, H avncrtoix.ro<;. B

a http://www.mathematica.gr/forum/viewtopic.php?f=40&t=522 92&start=20

A6aJJ.

A H

To Tpiyrovo AHK Eivm op0oyffivt0 Kat tcrocrKEAE<;, on6TE BE = AH = HK . Ta Tpiyrova B�E , KrH tx.ouv ..1E = rH , BE = KH Km �BE = rKH ( o�EiE<; ycoviE<; µE 7tArupE<; KU0ETE<;), 07tO'tE ea EtVat ro = <p ii ro + <p + 20 = 1 80° (Aµ<piBoA.11 nEpinTroCITJ tcr6•11m<; Tptyffivrov - 4o KptTJ1pto ). L'TJJV l 11 7tEpi7tTWCITJ 't(l Tpiyrova EtVat icra, 01tO'tE B..1 = rK . OWTEP11 L'TJJ nEptnTWCITJ Eivm ° e 90° + 28 = 180° => = 45 , 011A.aoiJ to ABr� Eivm 'TE'tpaycovo Kat to r Eivat µfoo toU AK 01tO'tE Kat naA.t Ar = I"'K -

----

ApKEi va &Ei�OUµE 6n Ar = rK . 'Ex.ouµE 6n EYKAEIAH.l:

B' 104 T.4/32

B' AYKEIOY '

Taf" :

An. KaKKa

EnavaArtnTI KE� A a Ktiat 1 �

AAfEBPA

KtlfnaKoxo'6Aotl K©vcnavriva

AO'KT)O'T) 1 11

'E«JT(J) 1tOA'll cOV'll J10 P(x) TtTOlO rocnt T O a0polaµa T(J)V (J'l)V'rtAt«JTIDV TO'll va tivm iao µt 6 Kat 1) oiaipt<Jl) TO'll µt TO x -4 vu oiVt\ tl1t0Ao\1t0 18. a) Na Ppt0ti TO tl1tOAo\1tO Tl)� Olaipt<Jl)� TO'll P(x) µt TO x2 -Sx+4. p) Na Ppt0ti TO 1tOA'll IDV'll J10 P(x) av tmxJ...tov yv©pi�o'll µt oTl txt• paeµo 3, o cna0tpo� TO'll Opo� \«JOVTat µt 6 Kat 1) apl0J11)TlK1\ Tlftll TO'll P(x) yla x= -1 l«JOVTat µt -12.

f(x) -3f(x -l)=Sf( -x)+50f( -x -2). (iii) Na A'll 0ti 1) t;ia©Gl) f(1)J1X -2)-�. 3x + 2 3Y = 171 (iv)Na A'll 0ti TO a6a'Tl)µa: E

{

t s x+ .

= 5 625� . _!_ SY ·

A Cf=:>f(-2)�25 =:>5-2a-4=5-2=:> -2a=2=:>a=-l. x. H f eivm yvricriros fia a= -1 dvm f(x)=5 au�oucra cruvaptT)crT). 'Etm -� �...fi =>f( -�) <f(�)5 <f(.../7). 5 f(x)-3f(x-1)=8f( ii)x _'Exouµe: -x)+5· 0f(x -x-2)<=> x x x x -l · . · -Z <=> 5 3 105 =8 5 +50 5 5 5 5 = � <=> 5 . 5x_5x <=>52x=52<=>-x=1 l . 11JJX - -1 iii) Eivm· f(n µx -2) 5 <=>5 2=5 <=>riµx= 1 <=>x=2K1t�,K 'Exowu::: { + � A\HJ1) : a)

p) i)

5 o(x)=x2 Enet8t1 o x+4tTJseivm OiatpEtTJs noA.urovuµo oeuttpou to un6A.oino paeµou, oiaipecrTJs tOUtTJsP(x)µop<J>µetltos u(x)=ax+p x2 -5x+4 eaµeeivm noA.urovuµo a, p ' 5 Enl1tl.£ov --4). Enaipvouµe: tm an6 tTJV x2tTJs-EuKA.tiOetas x+4=(x -l)(xoiaipecrf)s taut6tT)taP(x)=(x -l)(x --4)n(x)+ax+p (1). dvm icro Tµeo P(l) aepoicrµa trov cruvteA.ecrtrov tOU P(x) Km to U1tOAOl1tO tTJs omipecrf)s µe (x--4) ea eivm icrov µe P(4). Apa: {PP(4(l))==186 =:> { + { + { + {O·O·n(l)n(4)+a+P=6 a=4 =:>u ( x) =4x + 2. { + { =:> +4a+P = 18 P = 2 { { {199 A<pou to P(x) eivm noA.urovuµo 3°\J an6 paeµou tTJnoA.urovuµo V (1) cruµnepaivouµe 6n to n(x) dvm 1 paeµouµe oriA. aot1 Kat K =t:O. 5 r<x>= (i3 r+ G:r-1. n(x)=KX+A. K, A 'Exouµe P(x)=(x -l)(x --4)(KX+A.)+4x+2. f(2016) f(2017). 0 maeep6s6PQS tou P(x) eivm icros P(O), 01t6te: 5x + 12x=1 3 x. (-4)A.+2 =6) +4(-1) +2=-12 {PP(-1(0)=) =-16 2 {-l-2(-5)(-K+A. Eivm Oi cruvapti]cre� ( 5 )x ( 12 )x = x) ( A.=l ii , ( x) = ii eivm YVTJcriros => {K=2 =:>n(x)=2x+l=:> crt0 on6te Kat =:>P(x)=(x-l)(x--4)(2x+ 1)+4x+2=2x3 -9x2+7x+6. <p0ivoucres f ( x) = (i53 J c� J -1 dvm yvricriros <peivoucra f(x)=(sa+z t , x crto Enoµevros: 20165 <2017�f(2016)>f(2017). =t: 2. 5x+12x=13x (i3r+c:r -1 =0 f(x)=O. 1 -2, ) f. IlapatrJpouµe 6n f(2)=0, oriA.aot1 x0 =2 dvm 25 tTJs oev f(x)=OSa EXet Km aUri a<poupi�a.ri f(x) eivm rvricriros µov6t0VT) Au<Jl) : a)

E �.

-�

�

•

11:

3Y =

sx sx

171

5 x + i . 62 5 . .!. = 5 <=>

iv)

3x 3x 2 3Y = 1 7 1 2 3Y = 1 7 1 5x- y +3 = 5 1 <=> <=> 5 x + l 2 5 5 -y = 5 <=> 3x 2 3Y = 1 7 1 3 y- Z 2 3Y = 1 7 1 <=> <=> <=> x = y- 2 x = y- 2 4 3Y = 1 7 1 3Y = 3 Y = 4 SY

==>

<=>

•

<=> x = y - 2 <=> x = 2 .

x = y- 2

AGKT)GT) 311

E�

AivtTat 1) (J'l)Vllp'Tl)Gl)

a) Na (Jl)YJCpivtTt TO'll � apl0µo'6�

µe

Aua11 : a)

DF�.

A«JKT)O'T) 211

AivtTm 11 (J1)vapT1JG1J µt e � Km a a) Na Ppt0ti TO a, av tivm yv©aTO OT\ TO Gl)µtio A( avqKtl (JT1) ypaqnKl\ xapacna<Jl) Tl)� p) Av a= -1, TOTt: (i)Na O\aTa;tTt KUT ' av;otlaa «Jtlpa TO'll �

��),� i), f(../7). -

(ii) Nu Atl0ti 1) t;ia©Gl) :

lR .

ap•0µo'6�:

Kat

P) Na AV«Jt'Tt Tl)V t;ia©Gl) :

�

<=>

p«;a

:.\.O'KT) O'T) 411

AivtT at 1) (J'l)Vap'Tl)Gl) EYKAEIAHl: B ' 104 T.4/33

<=>

f(x)=311µ2 x+3 auv2 x -5.

Ma9f1JUl'TtKa yia 'TflV B ' Al>Ktio\l

u) Nu t;tTciO'tTt uv 11 f dvm cipTta ft ni:: ptTTft. p) Na l..:u 6d 11 1::;iaroa11 f(x)= -1. DF �. � Ai>O'll : u) 2 2 -x)=3'1 µ (- x) +3 <JtJv (- x) �. 2 2 3 TJ µ x + 3 <JtJv x 2 2 p) (::::> 311µ x+ 3 0-vv x (::::> 2 2 2 311µ x + 31-17µ x 3 217 µ x 2 2 2 311µ x - 1) = 0 311µ x+3=0 (:> ( 311µ x 2 2 it (::::> ( 311µ x = 3 � 311µ x it it (::::> ( (::::> ·

fta µ=3 Km A.-� civm f(x)=511µ(� nx). Eivm fta Kcl0€ x E tcrxQ€t 6tt A=25 -25 11µ2 G n�) =25 -2511µ2� =25(1-iiµ2 �) = x E E1ticr11<; f( -5 2; (1 + cruv :) = 2: ( 2 + �) = -5=f(x). A.pa 11 f civm apna. ) 25cruv = 2 ( ; -4=0 Eivm: f(x)= -1 (::> y) 'Exouµc: 5 1t 5 1t 51t 51t 51t -4=0 -4· -3)( B = 4 ( 21iµ-cruv- ) cruv-= 4ri µ-cruv-= 24 24 12 12 12 (::>(11 µ2x=l =1) 11µ2x=O) �11µx= 1 11µx=3 -1 11µx=O) 5 n cruv 5n ) = 2ri µ 5n = 2ri µ ( n- n ) = 2 ( 2 1iµ 2Kn , it x=2Kn+ n, it x=Kn, K E 12 12 6 6 1t 2· -1 =1 =2ri µ-= AiVETat 11 O't>VclPTll O'fl 6 2 f(x)=avv3(6n -x)+flp(i -x)·O'l>v2("i+x). µ -81t = ri µ -81t7_ = ri-�7Na unood;tTE OTt f(x)=O'\lVX. Na O'l>'YKPiVtTE TOt><;j aptOpoi><; r(zo�J Kat 2ri µ -41t7 � �) y) Na od;tTt OTt: O't>Vl5° -11p1 s0-v;. Eivm cruv(6n -x)=cruv( -x)=cruvx, 11µ(� -x)=cruvx Km cruv(� +x)= -11µx. A.pa f(x)=cruv3x+cruvx·11 µ2x=cruvx(cruv2x+11µ2x)=cruvx. H f(x)=cruvx civm yvricriro<; cp0ivoucra crto [O,�]. on6tc: O< < � >f( 5° 5° -11µ15°=cruv(45° -30°)-11 -30°)= cruvl (4 µ =auv45° · cruv30°+11µ45°· 11µ30° -11µ45°· cruv30° +T)µ30°· cruv45° - · · - · · (x

Z) .

AO'Kll(HI 5'1

u) p)

_ _

20 7

Afan1 :

-2!:_ 2017

AaKq cn1 6'1 :

-2!:_ 2016

2017

,,/2 '13 1 ,,/2 2

-2!:_) .

=:>f (-2!:_)

.!. 2

2016

,,/2 '13 I ,,/2 2

.!. ,,/2. 2

'E aTro 11 O'\lvcipTflO'fl f(x)=(p+2) ·11p(/...nx), onot> p,1... OtTtKoi npaypaTtKoi aptOpoi. u) Av 11 piytO"T11 Ttpft Tfl<;j f(x) dvm 5 Kat 11 paatKft ni:: piooo<; Tfl<;i dvm 6, va anood;i::Tt oTt n=J KUt J...=31 ' p) rta p=3 Kat va t>1tOAO'YlO'TEl 11 Ttpft Tfl<;j napciaTaa11<; A=25 -f y) Na t>1tO/...oytO'TOi>v Ot Ttpi<; T(l)V napaO'TclO'E(l)V: Sn 511' O'VV2 511' - qµ2 511' Kat 8=811p-O'l>V24 24 24 24 4 1t 1t 21t r = Scruv - cruv - cruv ,.

1..,-�

(

(�}

)

()) I;t Tpiyrovo ABr oivi::Tat oTt ot i::q>B, i::q>r dvm pi�E<; Tfl<;j i::;iaroari<;: 6x2 -5x+l=O. Na ppi::Od 11 yrovia A. Ai>ari : 1 I

{

a) 'Exouµc max f ( x) = µ + 2 =µ + 2 Km µ +2= 5 µ=3 21t 2 01t0t€: ' T =-=-, 2 => A. = -1 A.n A. -=6 A. 3

P) Av (xo,Yo) 11 AUO'fl TOt> O'\lO'TltJlaTO<;j yta TflV onoia tO"Xi>Et Xo<yo, va t>nol...oytO"Td 11 Ttpft Tfl<;j 11' , 2x 411' Y napaaTaari<;: K- ;0'\lv 8 - ; 0'\lv2� 1. 2 1 5 (L) Aual) :

a) fta va opi�ctm to cri>crtT)µ« npfact T Kat apKci x>O,· y>O. 6t€ exouµc: (.!')

{

(::::>

{

logx logy

( )

log � 2

1000

logx logy

� [Iog(xy) - log10 3 ]

{logx · (5 - logx)

6log10 =

(::::>

1 1

(::::>

{ Iogx · logy

6 logx + logy = 5

= 6 (::::> logx + logy = 5 log 2 x - Slogx + 6 = 0 logx = 2 , 11 (::::> logx + logy = 5 logx + logy = 5 logx = 2 , logx = 3 logx = 3 (::::> (::::> logy = 3 logy = 2 logx + logy = 5 X = 100 , X = 1000 ( , , A ) y = 1000 y = 100 1000 4 1l' Rp ) K-2 · 10 0

(::::>

{ { {

{

11 { u€Kt€<; nµc<; 11 { 1l'-'- 1 --- · cruv - --cruv2--. ·

EYKAEIAHI: B ' 104 'T.4/34

125

Ma6ttJ1«TtKa yta TtJV B ' AuKdoll

4D2+9+Dx2=4D -6Dx -1 (1) K«t D/ -4Dy+8D=O (2). a) Na ppt9o'6v ot Ttpti; Trov D,Dx,Dy .

=8cruv2�cruv2!!. - n1)+ 1 = -8 cruv2� 11µ2� + 1 = =l _2 11µ2� = �v2 _0· 8

p) Na At>9ti TO m>a'f11 p a:

A<JKTt<Jfl 811

AivtT«t oTt 'I 1tOAt>rovt>ptK11 (J\)VcJP'f11 <Jfl J1E a, P E IRl. P(x)=4x4+(a -l)x3+2Px2+(1 -P)x tlVCll apTta yta 'f11V 01toia l<JXVEl: P(l)= -1. a) Na 1tpoaowpiatn Tot>i; a,p fl) Na At>9tt fl aviaro(Jfl P(v. r::;. x) ::;; x3 +2x 2 -3x -l(l).

-7

•

q>ou 11 P(x) eivat aptta 0a tcrx;Uet: P(-1)=P(1 ), onote: {P(P(-t)=-11) =-1=i44-a.+a.-1++1+2132f3-1+1-+PP-7=-l -7 =-1 =>a.=P=l fta a= P=l eivm P(x)=4x4+2x2-7. fta va opi�etm 11 avicrro<Jll (1) npfaet Kat apKei x20. Tote txouµe: (1) <=> 4x2+2x-7::;; x3+2x2 -3x-l <=> 5x+ 5 0 <=> x3-2x20 <=>x3 -x2 -(x2+ x-6) 6� � x2(x-l)-(x+6)(x-l)�O<=>(x-l)(x2-x-6) �O <=> (x-l)(x-3)(x+2) + 0 <=> x E [ -2,l]U[3, � oo)� xE[O, 1 ]U[3,+oo). Auaf1 :

a) A

A<JKfl<J'I 911

; - ln(lne) -ln(Sve)= -i 17 "2 1 2logffi Iog(flp�)=loglS-l ii) log 4 2 fl ) Na At>9ti aviaro(Jfl : ln x+In-\+2<0 (1). y) Na At>9ti t;iaro<Jll : ( ) = ·2x (2) a) Na oti;tTt oTt: i) In 1

'I 6 -x 2 fil-"'10

a) i)

1 1 +.JilO

•

In� - ln(lne)-ln(5ve)= ln5 -lne -ln l ln5 -l21 ne=-1 --=21 --.2 . lo 17"2 2logffi�l 17"2 +log:;:;+l og(11µ-)=log--:og-:;-=2 17 17 17 -17 · -2 )= log-=2 =log(-· 15 =log-=logl5-log10=log15-1 . 10 fta Va EXet vo11µa 11 aVtcr(l)<J11 1tpfaet Kat apKei: x>O. Me x>O exouµe: (1) <=>ln2x-l nx3+2<0<=>ln2x-3lt nx+2<0<=> t {t2 - 3t 2 <=> { 1 t 2 <=> e<x<e2· 2( 2 ) = 22 +162"120 ·2x y) Eivm: (2) <=> ( <=> ( v'IO) =( v'IO)-22 2x <=> <=> ( v'IO) . ( v'IO) 24-x= 1 <=> <=> (2v'TI 2 v'IO) =l<=>x=4. Aua11 :

11) � 4 '12

'12

f})

lnx

6x "12 +� 6-x m+ 6 -x m+ 4 -x <

-4

+ A<JKfl<J11 1 011 l:;f tva m)<J'f11 pa OVO ypapplKIDV t;taroatrov J1E ayvroaTot>i; x KCll y l<JXVO\lV Ol <JXEaE�:

{

5J.x+Ay : 5 2x-1 6y 5x . SAY

= 62 5 I; ( 52-2y

)

!Rl.. (1)::::> 4D2-41 D+ l +Dx:2+6DAEx+9=0::::> (2D -1)2 + (Dx+3)2=0 ::::} D-2 Kat Dx= -3. E�allou: (2)::::> D/ -4Dy+4=0::::>(DY -2) =0::::>Dy=2. Aq>ou D=FO, to <JUcrtll µa exei µovaotKfi AU<Jll tllV Kat y - = 4 o,yo) l'""' - 2)x+0..+ 16)y 54 { 50..5x+(A+2)y 52 { -

yta T� otaq>opti; Ttpti; 't'11 i; 1tapapfapot> Auari :

a) 'Exouµe:

(I)

1 1 i:- XO� D

fl) (I:) <=>

(.A 2)x + (.A + 1 6)y = <=> x + (.A + 2)y = 2 'Exouµe·. D= "- -1 2 "-A.+ 1 6 +2 4 ii. + 16 =(/.. 5)(/..+4 ) D =

•

A.2 J.. 2 0= -

4 =2/.. 2 4 D I ii. 2 il. + 2 1 2 =2A -8 . Av tote 0 aoii av crU<Jtllµa exet µovaotKfi DAU<Jll tllV (Xo,Yo), 01tOU Dx 2 A.- 2 4 - Dy - A. - 8 Av A Kat xo- D - A.-s) o Y (A.+4) ( (A.-s) (A +4) ' tote Kat apa to (I:) eivm aouvato. Av tote Kat apa to (I:) eivm -

A.e !Rl.- {-4,5 }, ro 2 =-4 D=O A.=5 D, x= -32::FO, aouvato. D=O Dx=-14=FO, D=FO 111..

A <JKfl<J11 1 1 11 AivtT«t 'I (Jt>VclP'f11 <Jfl f(x)= logllog(x -5)1. a) Na ppt9ti TO 1t£Oio optapot> 't'11 i; f. J1E Tot>i; fl) Na ppt9ot>v Ta (Jflptia TOplii; 't'11 i; a;ovti;. y) Na ppt9ti o pa9poi; Tot> 1tOAt>rov'6pot> P(x)=f(A)x5-(lf(A)-41)x3+2x+l yta otaq>opti; Ttpti; Tot> A

Ea)!Rl.r. ta va op�etm 11 f npfaet Km apKei: log(x (x> -5)5=FO), (x0111..aoii (x>5 Km x 5::F+oo).5>0 Kat x::F6). Apa DF(5 , 6 )U(6, l), KatKatteAtKa Eneiliii to 0 oev avi]Ket ITTO 1te0l0 optcrµou tllc; <ruVclPtll<Jllc;, 11 Cr oev teµvet tOV y 'y. H Cr teµvet tOV 'x Kat µovo av 11 �icrro<Jll x'Exouµe: f(x)=O exet AU<Jll . 5 f(x)=O<=> llog(x- )l= l<=>log(x -5)=1-5 1 ii 5 ii x 10). log(x-5)=-l<=>(x-5=10 ii x-5�)<=>(x=l 10 Onote 11 Crteµvet tov x ' x ma K(l 5 , 0) Kat A(��,0). y) Av f(A.)=FO, 0111..aoii av 1 1 5 5 AE ( 5 , 10 ) u ( 10 , 6 )U(6,1 5 ) U(l 5 ,+ oo ) tote 0 pa0µoc; tou P(x) eivm icroc; µe 5 . Av f(A.)=O, tote A.= 1 5 ii A.-��· Km crttc; ouo nepumocretc; eivm P(x)=-4x3+2x+ 1, 0111..aoii np0Ke1tm yta noA.urovuµo 3°u pa0µou. At>ari :

EYKAEIAHl: B ' 104 T.4/35

Ma011paTt.Ka yia 'f'IV B' A1>Kti01>

EnavaAf1trTI KC� AaKtia&I � r&WIJ&Tpia�

Xpl]o"to� n. Taup 0.K1)�

AcpIBproµtvo crniv µviJµri tou aymtritou cpiAou 0roµa P mKocp-rcraA.ri H f:1ttAOyTJ 'f(l)V acrid)O"f:(l)V eytvf: ano to j31j3A.io «feroµeTpia B ' AUKf:lOU» toU A . I. :EKtaOa (1983) Ka0ro<; apKete<; ano au-re<; npocrcpepov-rm Km y1a µta Oel>TepT) touA.ax1crtov avnµe-rromcrri . /\<JK11 <Jtt 1 11 :

Kr => . zr · E>A KM 1 => zr = -1 · -· = ZA E>M Kr ZA 2 KM EB + zr = .!. . KB + Kr = .!. . 2KM = 1 EA ZA 2 KM 2 KM AB + Ar = J . KB + Kr AE AZ ' Acpou:' M µecrov wu Br KM = ---Ai><Jtt : <l>epvouµe niv 2 BH I /EE>Z Km niv rA I IEE>Z To-re ano 0eropriµa 0aA.ft exouµe a, AB Ar AM 3 Av EZ//Bf, toTf: Ar AA AB AH , vncrto1xa = AE AZ AE> 2 Km = AE AE> AZ AE> /\<JKt} <J'I 211 : AivtTut TtTpaycovo ABrA 1t#.£vpa� AivtTut Tpiycovo ABr Kut 0 1'0 Pu pi>KtvTpo Tov. A1to 1'0 0 <ptpvovµt tvOdu (t) 1t01> U<pl]Vtl Tl� KOp1><pE� B Kttl r OTO lOlO 11J1lt1tl1tt00 Kut Ttµvti T� 1tA&upt� AB Kut Ar OTu <n)µdu E Kut Z uVTiaTotxu. Nu oti;tn 01't

�

•

- = - = -- = - .

u. A1to T11V Kopv<pl] A <ptpvovµt tvOda (t) 1t01> 1'tµvtt T"lv 1tl.&upa Ar OTo z Kut T"lv 1tpotKTU<n) T"l� Br OTO E. Na od;tn OT\

� + � = -;.. . AE

fvropi�ouµe on O"f: Ka0e op0oyc0v10 -rpiyrovo 1 1 1 , ABf µe U1t0Tf:1VOUO"U Bf 10"'.Xll, f:l: -2 +-2 -2 AU<Jt} :

Ilpocr0faovm<; Kma µeA.ri

AB + Ar = AH + AA = AH + AA AE AZ AE> AE> AE> -

----

Ta -rpiyrova BHM Km fMA eivm icra (y1mi;) onoTf:

HM = MA . AH + AA = ( AM - HM ) + ( AM + MA ) =

= 2AM = 2 . l AE> = 3AE> . 2 AB + Ar = -3AE> = 3 , Apa . AE AZ AE>

U CL

•

ApKei A.01nov va Oei�ouµe on unapxe1 op0oyc0v10 -rpiyrovo µe U\j/O<; AB=a Km Ka0f:Te<; nA.supe<; icre<; µe AE, AZ. Av OT)AUOyt T) Ka0e'tT) crniv AE O"TO A TEµVel niv Bf O"TO N, TO'Tf: apKei va Oei�ouµe on AZ=AN . Ilpayµan Ta -rpiyrova ANB Km A/!:J.Z eivm icra (EA

xouv AB = M = a , B = � = 90° , N AB = Z A � o�eie<; yrovie<; µe nA.eupe<; Ka0f:Te<;). Apa

AN = AZ .

B ' Tp07t0�

s: ;�

ApKel, va uf:..._,ouµe on ,

AE + EB + AZ + Zr = 3 , AZ AE ·

, EB + zr = l . TI EA ZA

•

Av 01 EZ, Bf -reµvoVTm crto K TOTf: crl>µcprova µe w 0eropriµa MeveA.aou cr-ra -rpiyrova ABM Km

EB · E>A · KM = 1 => , ArM exouµe: EA E>M KB EB · 2 · KM = 1 => EB = -1 · KB Km EA KB EA 2 KM -

/\aK11 <Jtt 3 11 : AivtTut 1'npaycovo ABrA qyqpu µtvo (Jt Ki>.U..0 A1t0 Tl� KOp1><pt� A, B, r,

( O,R ) .

A <ptpvo'Uµt Tt� AA ' , BB ' , , KaOtTt� (Jt µiu t<pU1tTOJ1tvll TO'U Ki>.U..01>. Nu od;t1't

rr' M' (S I ) OTl AA'· ff'+ BB '· M' = R2

Ano to Ktv-rpo tou Kl>tlou 0 cpepvouµe niv su0eia (o) napnAAT)AT) crniv ecpamoµtvT) (e) Km fo-rro N, :E, A Km H Ta crri µeia toµi}<; 'tT)<; µe T�

Al><Jtt :

EYKAEIAm B ' 104 T.4/36

AA' ,

Ma011panKcl yia Tt)V B ' A1>Ktio1>

BB', !:!.!:!.. ' Km rr' avticnoixa. Tote exouµe:

AA '· rr · = ( R AN ) ( R + rH ) (1). -

tva op1aptvo (O'Ta8tpo) O'l}JltlO p O'TO &O'O>T&plKO Ano TO p q>tpvoupt TIJ:Xaia tu8tia (t) 1t01> Ttpvtt TflV nl.tupa Ox aTo 0'11 Jl&io B Kat TflV nA&upa Oy O'TO O'l}Jl&io r. Na Ot\;&T& OTl

TTI�·

(0)

�

Ta tpiyrova ANO Kat /\

�

2 AE 2 , rr ' = rE . (Anocrtac:rri 2R 2R

'Exouµe: AA ' =

Ox.

(E)

OB = Or = R , L = H = 9 0° , H O r = r B O O�SIB� yrovie� µE 1tAeupe� Ku0sts�) upa B r = OH . 'Etcrt rH 2 + BL 2 = rH 2 + OH 2 = or 2 = R 2 Apa AA '· rr '+ BB '· �� ' = 2R 2 - R 2 = R 2 . B ' tpo1to�

Ano to c:rri µEio <pepvouµs tt� Ku0sts� PE Km crtt� nA.eupe� Km Oy avticrtoixa Km ano to c:rri µsio r Tl'IV rH KU0ST11 O'Tl'IV Tots: Auaq :

OfH sivm icra (ytati exouv

B�O Kat OHf sivm icra (ytati exouv /\

P Ox

•

(E)

N = H = 9 0° , N O A = H O r , OA = Or = R ) 01t0tS AN = rH Kat tots AA '· rr ' = R 2 - rH 2 (2). Dµota �picrKouµe ott BB '· M ' = R 2 - BL 2 (3). Ilpocr0farovta� KatU µEATt tt (2),(3) EXOUµE AA '· rr '+ BB '· �d' = 2R 2 - ( rH 2 + Br 2 ) . Ta tpiyrova

1 = cna0tpo + ( OBP ) ( orp )

c:rri µsiou ritlou mto xopofi ft s<pantoµEvri tou. rsroµetpia OtKOVOµtKOU ritlou, T . KaA.onic:rri - r. TacrcronouA.ou crsA. 279, eKooc:rri 1 977).

1 - ( OBP ) + ( OrP ) 1 ( OBP ) ( OrP ) ( OBP ) · ( OrP )

--- + =

f ·9B·rH (OBr) (OBP)·(OfP) 1 · 9If ·PE · � · Of ·PZ =

rH 2rH - = ---- = = ---PE · Or · PZ ! . PE · Or · PZ 2 = 2 · 1 · rH . A<pou, to c:rri µsto stvm crtaPE · PZ or

0spo, exouµs Ott PE Kat

PZ sivmrHyvrocrtu µiJKTt Kat /\

' OfH SXOUµS ' Ott' Or = l']µ 0 (yvro1 1 spo . O'Tll, ) . npa ( OBP ) ( orp ) O'to tptyrovo 'A

<Ha 0

B ' Tpono�: 'Exouµs:

( OBr) 1 --- + 1 ( OBP ) ( orP ) ( OBP ) . ( OrP ) -1 OB · or · l']µ(co + <p) =2 - �·------ = 1- OB . OP • l']µco . -1 or . OP · l']µ<p 2 2 2n· µ ( CO + <p) = crta0spo, . OP 2 l']µco · l']µ<p -

) (E� ..e.____.i.....i.__..;. .. ....�� B' A' E ll' r

Apa M'- IT' =

( ��IBJ (2��J =

= EA'

'Oµota BB '· rr = EK 2 Apa I

•

AA'· rr '+ BB '· rr · = EA2 + EK 2 = 0E2 = R2 •

A O'Klf O'lf

411 :

0tropoi>pt KUpn} yrovia xOy Kat

---

EYKAEIAlll: B' 104 t.4/37

Ma&qµa'T1Ka y1a ntv B' At1Ktiot1

AaK11a11 5 11 : 0tropoi>µt Tpiyrovo ABr, T'lV 016:

µtao AM KU\ T'lV 01x0Toµo T01> AA. 0 ntptyt ypaµµtvo<; Ki>KA.o<; O'TO Tpiyrovo AAM TtµvE\ T'lV 1tAEvpa AB O'TO E KU\ T'lV Ar O'TO z. Na od;ETE OT\ TU Tpiyrova BAE KU\ rAZ ElVU\ \O'OOi>vaµa, 011Mio1l : =

(BdE) (rdZ) .

R2 2

€xouµe KA 2 + K0 2 = 2U 2 + - =>

2 K0 2 = 2u 2 + R - KA 2 . 2

Ilpo<pa.vro� ta. U'l'TI �H, �e trov tpiyrovrov �BE, �rz eivm icm (t<ho'tT)ta. oixotoµou) onote a.pKei va. oe�ouµe ott BE=rz. Ilpayµa.tt a.no ou va.µTI CTT) µeiou co� npo� K'6tlo €xouµe:

Ava11 :

B� · BM ( 1 ) BE · BA = B� · BM => BE = BA r� Km rz . rA = rM . r� => rz = - . rM (2). rA

Alla a.no to 0effipT1µa. ecrrotepucft� oixotoµou €

xouµe:

B� r� - = - (3). BA rA

ApKei A.omov va. oei;ouµe ott

2 R2 2�2 + = R2 - 2KA2 TI, R = u 2 + KA 2 ' 2 4 ' A A2 2 2 ' + KA , nou tcrXUet. TI l"\.L.l = '

v A 1".L.l

AaK11a11 7'1 : 0tropoi>µE tva nupull11Mypuµµo

ABrA µE tµpaoo K 2 :El>votol>µt To µtaov T'l<; Ka8t 1tAEvpa<; µt Tl<; U1tfvUVT\ KOp1><pE<; 01tOTE O'Xl)µ«Tl�ETE tva OKTa"(<OVO. Na 1>1tOAo"flaETE TO tµpaoov T01> OKTU"(cOV01> Ul>TOi> O'l>V«PTllO'E\ T01> Eµpaooi> T01> ABrA ( •

H I::.

A1to ( 1 ),(2),(3) npoK'6ntet BE = rz .

AO'KflO"l 6 11 : Tpiyrovo ABr ElVU\ E"("(E"(p«µµtvo

at Ki>KA.o (O,R). H «KTiva OA 01xoToµdTUl ano /\

K2 ).

Ava11 : Ot KM

1 3

T'lV 1tAEvpa Br. Na od;tTE OT\ E<p B . E<p r = -

Km NA oiEpovtm a.no to KEvtpo 0 toU na.pa.llT1A.oypaµµou Km OlXOtoµouvtm.To

A�AN eivm na.pa.A/µµo, apa. OZ = ZK =

OK (1 ) . 2

:Eto tpiyrovo ONK to E eivm to pa.puKevtpo, apa.

OE = _!_ OA (2). ApxiKa a.nooetKV'6etm ott A > 90° . IIpay 3 µa.tt a.<pou ta. A, O eivm eKa.t€pro0ev 'tTJ� Bf ea. ei( OZE ) OE · OZ 1 - , , = = (3). vm BAr < BAB =18D° onote Etcrt exouµe ( OAK ) OA . OK 6 3&' -M'> 1ro> =>A>W' . (Me PO.<JrJ avnj r17v irapanjp17<JrJ <ITO revxo<; 1 03 - Alla ( OAK ) = _!_ ( ONAK ) = _!_ _!_ ( ABMK ) = Ava11 :

I'swµerpia B AvKeiov-<56817Ke µza rp1yovwµerp1K� AV<Jr/. llapa8i:rovµe <IT11 avvexeia KW µza Ka8ap0. yswµerpzK� AV<JrJ)

KA ' , ecpr = KA , onote: Kr KB KA2 ecpB · ecpr = . ApKei A.omov va. oei;ouµe KB · Kr ott: KB · Kr = 3KA 2 , ii R 2 - OK 2 = 3KA 2

'Exouµe: ecpB =

(Mva.µTI CTT) µeiou co� npo� K'6tlo), ii

OK 2 = R 2 - 3KA2 .

Ano to 1° 0effipTlµa. oia.µforov crto tpiyrovo AKO

2 2 •

_!_ _!_ · _!_( ABr � ) = _!_ · K 2 . 2 2 2 8 ·

1 Apa. ( OZE ) = - K2 . Dµoia. ppicrKouµe ( CYZII) , 48

EYKAEIAHI: B' 104 'T.4/38

Ma911po:ttKa yia TflV B ' Al>Ktiol>

( OHE> ) . . . KA1t. Km teA.uca EXOUµ.£ on ( EZHE>I <l>PXE ) = 8 · -1 . K2 = .!. . K2 • 48

A. a K11 cn1

811 : AivtTm Tpiyrovo ABr £'Y'Y£YPaµ.µ.tvo

CJ£ rildo

( 0, R)

Km BH =

R.J2

TO V'f'O� Tot>.

Ano TO H <ptpot>µ.£ HA l. AB Ka\ HE l. Br Na od;tT£ OT\ OB l. AE Ka\ OT\ Ta CJtt pda A, O,E dvm (Jt)Vtt>OtiaKa. •

At>a11 :

Bi\ . BA = BH2 = BE . Br => ArEL'.'.\ eyypa'lftµo, => A = E 1 . NJ..a A = B 1 (ycovia U1t0 xopot1i; /\

·/\

ocooeKaycovo exouµe Ott A OJ\ = .... =J\2 0A =3if . 'Etcrt crto op0oyrovto tpiycovo A10B 1 txouµe ott

A1 B1 =

OAI R Rf3 . -= - Km OB1 = R · cruv30 = -2 2 2 0

'Oµota O'tO op0oyrovto tpiycovo Bz OB I exouµe OB 3R OB B I B 2 = 1 = R f3 B B3 = 2 = . . . KA.1t . 2 2 8 2 4

Km ecpanttoµEvrii;).

B B 3 + ... = 2 R + J3 + = .... = +i 2 l 2 4 ····

'Etcrt txouµe S = A 1 B 1 B 1 B

R + R J3 + 3R + 2 4 8

Apa E 1 = B 1 => L'.'.\E I l(t) => L'.'.\E l. OB crto K acpou (t) l. OA . ApKei nA.Eov va eivm BK=BO. Ta tpiycova BL'.'.\E Km BAr eivm oµota, onote

BK = Bi\ . AA.A.a: BH Br BH2 Bi\ - = BA = BH2 = BH2 = BH Br Br BA · Br 2R · BH 2R BK = BH => = BH2 ( R� r = R = A a BK BO BH 2R 2R 2R

(:Ero reiJxo� 1 03 -I'ewµt:rpia B Ama:iov- mr'ffpxe Km eva� &vrepo� rp fnro � AV<Yff � r17�) . .AaKttatt

9 11 :

AivtTm Kavov1Ko orootKayrovo £"f'Ytypaµ.µ.tvo CJ£ Kt>ldo 0,

(

]

To a0potcrµa µfoa O"TTJ napev0e0'11 eivm a0potcrµa aneipcov opcov cp0ivoucrai; yecoµ£tptKt1i; npoo8ou µe fj = Myo /... =

Apa

�·

f\ a K 11 a11 1 011 :

(L �J. 1 - /...

( J3 ) .

l = R· 2+ fj

1-2

Aiv£Tm T£Tpayrovo ABrA nl..et> pa� a Ka\ 0 TO KivTpo TOO. Mt KivTpo T� KOpt><p� A,B,r,A Km aKTiva OA=OB=Or=OA <ptpvot>µ.£ oiaoox1Ka T£TaPTorildm not> Tiµ.vovv T� nl..et> p� AB, Br, rA, AA CJTa att µ.tia A, 0, z, M, K, E, H, I. Na t>no/..oyiCJ£T£ TTtV ntpiµ.tTpo Km To tµ.paoov TOO CJTat>pot> HEOKMOZ00AIOH not> <JX11 JlaTi �£Tm.

Ar = a Ji H 8tayrovioi; Ar2 = 2 · a 2 Km aKtiva tcov tetapt0KUKA.icov eivm , ta tµT)µata , �E, �H, Al, AA, R = a · Ji , onote 2

Aua11 :

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A 1 A 2 A 3 A 12 ( R) . 'ECJTro B 1 11 npopo!..1] TOt> A 1 CJTllV OA 2 , B 2 11 npopo!..1] Tot> B 1 CJTTt V OA3 , B3 11 npopo!..1] Tot> eivm icra µ£ B 2 CJTTtV OA4 , Km ot>Tro Ka0 ' t;l]�. Na ppdTt Be, BZ, fK, fM TO a9pOlCJJla a - R = a - a!!- = � · ( 2 - Ji ) Kat ta tµt1µas = A I B I + B 1 B 2 + B 2 B3 + ... (A.E.I. 1964) ••.•

EYKAEIAHI:. B ' 104 T.4/39

m EH = IA=

EH = fiE

Ma011pa'nKO. yia TI}V B' AllKEio\J

-1) = fn af-a.{./2-1) =

E>Z = MK £xouv µftKo�

J2 = � ( 2 J2 ) J2 = a ( J2 -

'Erm ri m::piµe-rpo� wu cnaupou tcrou-rm µe

IT = 2nR + 4 · EH

= a { nJ2 - J2 + 1 ) . ·

A-B

1:µpaoov Tov K'VKAtKou Tµ1lµaTOc; Auari : i) <l>f.pvouµe 'tO U\j/O� rM CHO tcr6nA.eupo

AB = ' ABf . TO'tB ' AM = l 'tptyffiVO

a J3 = ( J6 - J2 ) . J3 rM = 2 2

•

J6 -2 J2 Kat

H Mf dvm µecroKa9ew� 'tl'J� xop8ft� 8tf.pxe-rm an6 w Kf.v-rpo wu K'6KA.ou 0.

AB,

apa

fm w eµpaoov £xouµe:

( fiEEoZsfi ) = ( firB) - ( K'toµr, Ei ) = a2

- -

· -

A.pa £xouµe

( 4 rt ) 2 8 8 E = 4 ( fiEOOZBfi ) 4 ( fiEH) =

- -

· -

- ·

2 J2 ) a2 ( 4 = 4 ( 4 rt ) 4 = -g 2 a2 a2 a2 = 2 ( 4 n) 2 ( 6- ifi ) = 2 ( W. -n-2) . 1

AO'Kl'JO'l'J 1 l '1 :l:TO napaKaTro ax1lµa dvm

AB = Br = rA = Or = J6 - J2 . B

'E-rcrt O"'tO opeoyc.Ovto -rpiyffiVO OMA an6 rrue . E>eropriµa £xouµe: OA 2 = R 2 = AM 2 + OM 2 =

AM 2 + (Or + rM) 2 = AM 2 + or 2 + 2or . rM + rM 2 = Ar 2 + or 2 + 2or . rM = 2or 2 + 2or . rM = 2or (or + rM ) = 2or . 0M .

J6 ) ( ( -; J3 } l\pn R' = 2 J6 - h {J6 -h +

{1 �) = ("6-h)' (2

2 ( v'6-h )' + A.pa R = 2 .

ii)

+ ,/3 ) = 4

0 KUtltK6� wµfo� 0 , AB av-rtcr-rotxei cre -r6�o 3 0° . 'Ernt w eµpaoov wu KUtltKou -rµftµaw� AB

(

)

(

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rt · R 2 . 3 00 _!_ , R 2 · 11µ300 360 2

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)

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Ta�":

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AOK'IO'I 1 .

Na ppdTE T� t;1aroat� TO>v Ki>tlrov no\l OtipX,OVT«l U1t0 TO A(-1,0) K«l E<pcl1tTOVT«l T CoV t1>0ElcOV SI : 2x - y - 2 = 0 K«l 8 2 : 2 x - y + 1 8 = 0 .

Ilpocpavroi;; 01 cu0Eici;; E1 , E 2 Eivm napUAATJAEi;;, Kata <JUVEnEla 'I aKtiva cv6i;; K'l'.>tlou

Aua11 :

1tOU E<pU1ttEtCll cmi;; E1 ' E 2 Eivm p _!. d( E1 ' E 2 ) . 'Ecmo OT)µ£io tTJi;; E1 . T6tE

=2 A(x0,y0) 2xo -yo + 181 j2 +1 8I =4J5 . d(EpE2 ) = d(A,E2 ) l �i +(-1)2 J5 Apa aKtiva trov �TJtouµcvrov K'l'.>tlrov Eivm JS = 2 . To K( x0, y0) Eivm Kevtpo cv6i;; an6 to0i;; �TJtouµcvoui;; K'l'.>tloui;; av Km µ6vov av (AK)= d(K, E1 ) = d(K, E2 ) 'ExouµE d(K, E1 ) = d(K, E 2 ) -21 = l 2x0 -y0 + 181 l 2x0 -y0 JS JS 2x0 -y0 -2 = 2x0 -y0 + 18 t1 2x0 -y0 -2 = = -2x0 + Yo -18 4x0 -2y 0 + 18-2 = 0 2x0 - Yo + 8 = 0 y0 = 2x0 + 8 EniOT)i;; AK = d(K, E1 ) <::::> �(Xo +1) 2 + Yo 2 =2JS (x0 + 1)2 + y02 =20 <::::> (x0 + 1)2 +(2x0 + 8)2 =20 <::::> x0 2 +2x0 + 1 +4x02 +32x0 + 64=20<::::> 5x02 +34x0 +45 =0 <::::> x0 =-5 t1 x0 =- 25 fta Xo = -5 exouµE Yo = -2 KCll yta Xo = _2 5 , ,cxouµE 0 = 22 . n.pa unapxouv , uuo, KUIV\,01 µ£ S ( S9 )2 + (y - S22 )2 = 20 KCll E�lO'cOO'Eli;; X + (x+ 5 )2 +(y+2)2 =20 p

<=>

AOK1)0'1 2 .

' A

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Aivamn 'I nupapol11' y 2 = 4 x K«t 'I t1>0da y = A(x - 1), A '* 0 'I 01t0la Tif1VEl 'f'IV mxpapo/..fa OT« <J'lf1El« A,B. Na od;ETE OTl Ol E<pU1tTOf1EVE�

't'I� nupupol..1\ � OT« <J'lf1El« A,B Ttf1vovTm Kcl0ET«K«l 1tclV(I) OT'll Ott1>0ETOi>O'«.

01 ouvtEtayµevci;; trov OTJµ£irov toµili;; A,B Eivm 01 Mcrc1i;; tou oucrtilµatoi;;

2 = 4x {yy=A(x-1)

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(1)

(2)

H (2) µc tTJ Poi10cta tTJi;; (1) ypacpEtm

y=A (� - 1 ) 4y=Ay2 -4A Ay2 -4y-4A=0 (3) 01 tEtayµevci;; toui;; Yi . Y 2 Eivm Mcrc1i;; tTJi;; (3) , 01tOtE y, . Y 2 = -4 01 E�lO'cOO'Eti;; t(l)V E<pantoµevrov E1 : yy1 =2(x + 1) } , crta ElVCll E 2 : yy 2 =2(x+x 2 ) Km exouv <JUvtEAEO'tEi;; 81EU0uvOT)i;; A, = 2 KCll 1.,2 = 2 avt1crtoixroi;;. Apa A1 A 2 = Yi Y 2 =-1=> E1 .l c 2 4 fta va teµvovtm 01 E1 , E 2 npenE1 Km apKEi to {yyyy1 == 2x2x ++ 2x12x (L) va EXEl AUOTJ O'UcrtlJµa 2 2 'ExouµE: ( ) { yy, = 2x + 2x1 ( 4) aUa: L y(y2 -y1)=2x2 -2x 1 ( 5) ( 5 )<=>y(y2 - Y1 )= Y{ _ Y�2 <::::> y= Y1 � Y2 , acpou y1 y2 (Ola<popEtlKU 0a EtXClµ£ X1 = 42 = 42 = X2 OTJAaoil , npayµa atono ). Enoµevroi;; 2 ( 4) <::::> 21._2 + Yi2Y2 = 2x + 2x1 <::::> 2x = -2 x =-1 Apa E1 , E 2 tEµvovtm crto OTJµ£io (-1, y, � y2 ) to onoio avilKEl crtlJ 81su0Etoucra 8 : x = -1 <=>

<=>

A,B

2l.

<=>

21:. 2L_

<=>

'• OKTIO'I 3 . .-t.

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EYKAEIAlll:. B ' 104 T.4/41

x 2 = 2py Iax-6.:i To ioio x 3 = 2py ; •

yu1

Ma91JJUlTtKa yia TflV B ' A1>K£lo1>

TflV K«JUWATJ

To cµpaoov rou tptyrovou OAB civm

A(x1, y1) B(x 2 , y2 ) r(x3 , y3 ) avflKouv CJTilV napapoA.fi x 2 = 2 y Km civm oiaooxiKe� Kopucpe� napaU11A.oypaµµou. = 21 2 4x 2 1- y 2 2 4x 2 - y 2 2 Tote ta tµfiµata Ar,B� exouv KOlVO µfoo. Ano auto nporintct avc�apT11t0 rou <JT)µciou M (xp y1 ) . {y,X1 ++ yx3 == yX2 ++ yx4 =S (2){l) Eni<JT)� 3 2 42 + --Eni<JT)� exouµc X; = 2 y yia Ka0c i= l, 2, 3, 4. + x8 4 x 1 - 2 y1 4x 1 + 2 y1 2 2 Apa ( 2) x1 2 + x/ = x/ + x/ civm Oµoiro� ppicrKouµc y A + y B = y apa ( x , + x3 )2 -2x1x3 = ( x 2 + x 4 )2 -2x 2 x 4 � 2 µfoov rou AB . x1x3 = x 2 x 4 = P (3) Na ppc0ouv Ol Kopucpe� r, � tctpayrovou ABr� A6yro trov (1), ( 3) oi api0µoi (x1 ,x 3 ) Ka0ro� Km otav A(2, -1) KmB(5, 3 ). 1')0t ( x 2 , X 4 ) civm pi�c� Tll � c�icrrocrct� A-601) : 'Exouµc: 2t - St + P = 0 . Apa (x,X ==Xx2 ) Ti (xX, ==Xx4 ) , onotc AB=(5-2, 3+1)= (3,4), onotc I ABl = ·h2 +42 =5. AA.A.a: sr .l AB � Br =A.· (-4, 3 ) Km j B � = I AB I 3 4 3 2 ta <JT)µcia A, B cruµninrouv Ti ta <JT)µcia A, � � I A, I · 5 =5 � I A, I =l �A.=±1. cruµninrouv, npayµa arono. rta TllV Kaµm)A,1') X 3 = 2 y OCV tcrxf>ct tO tOlO. Ilpayµan av 0cropficrouµc ouo <JT)µcia Tll � Kaµm>A.11� A(x" y1) B(x 2 , y2 ) tote ta cruµµctptKa tOU� r(-x1,- y 1) �(-x 2 ,- y 2 ) 00� npo� ro 0(0, 0) avflKouv crTilv Kaµm>A.11, acpou EnaA.110cl>ouv T11V c�icrrocrfi Tll � · To ABr� oµro� civm napill11Mypaµµo, OlO'tt Ol otayroviE� tOU . oixotoµouvtm. B� A'6a11 :

Yno0etouµc on ta <JT)µcia

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(I)

'tO M

B(S,3)

Ar,

A<JK'l<JT) 4.

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\ \

__ ,,,,.. ,,,,..

Av A.= l, tote: Bf=(-4,3 ) � 'Ecrtro M(x" y 1 ) ruxov <JT)µcio Tll � X r -X B : -4} X r : -4 + x � = -4 �5 = l } . uncppoA.fi� c :4x 2 - y 2 = I . H c�icrro<JT) Tll � Yr - yB - 3 Yr - 3 + y - 3 + 3 - 6 c<pantoµ£vri� crro M(x" y 1) civm 4xx1 -yy 1 = 1 Km autfi teµvct n� acrl>µntrotc� y = 2 x Km Av A.=-1, tote: y = -2x crta <JT)µcia A( 4x1 -l 2y1 , 2x1 l- y1 ) Km Br=(4,-3) � = 4 + x 8 = 9 } = -3 + y 8 = 0 A'6a11 :

�

EvrcA.ro� oµota unoA.oyi�ovtm oi cruvtctayµevc� tOU �.

EYKAEIAHI:. B � 104 T.4/42

YnE'UOt>voi Ta1;11 � : ti.

r ' AYKEIOY

ApyuQ<i!,CTlc;, N. Avrcov6nou/..,o<;, K. BaKa.A.6nou/..,o<;, I. AouQii>a<;

AaKt1at1 � TI10avoTt1Twv TO>V Baail.;q KapKav11, '1>payKiGKO'll Mm:paiµ11, Ilavayt<l>'f1l Avopt01tO'llAO'll AGK'lG'l I 'l

Avo <potT11Ti:i; Oa nai;ovv i:va online video game napall11J..a, Km G'll fl<provovv v1K11ti)i; va dvm tKdvoi; 1tO'll npcl>Toi; Oa Ktpoiatt Tpta nmxviota µt 'f1lV npovno9t<J11 OTt otv Oa 1tai;ovv 1t0.vro ano 4 nmxviota. I:t K0.9t O.ll11 1ttpi1tTO><J11 Oa t>1tap;1:1 iaonal..ia. Av a dvm TO anoTi:l..taµa va Ktpoiatt o npcl>Toi; <pOt'f1lti)i; i:va nmxviot Kat p dvm TO anoTi:l..taµa va Ktpoiatt o OEVTtpoi; <po1T11nti; i:va nmxvioi, va ypa<pd o 01:1yµaTtKoi; xropoi; TO'll 1tttp0.µaToi; µt T1l pol)Otta Otvopootaypaµµawi;. Ilota dvm 11 m9aVOT11T« iaonal..iai;;

Aua11

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N(r) = � .!.. P (r) = N(n ) 8 4 w

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A GK'l Gfl 2 'l

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EYKAEIAH:E B' 104 T.4/43

Ma011J1«T1K6. 'Y•« TflV B · AllKEio\J

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a) To oevopoouiypaµµa nou Ka0opiset TIJ cretpa na.Aat6TI)T(l(; Km w XPffiµa eivm w aK6A.ou0o:

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P(Z) =

5 9

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P (I) = P( (B - r) U (r - B ) ) = P ( B -r) + P(r-B ) = P( B ) + P(r) - 2P ( Bnr )

a) Q OEtyµanKO� XOOPO� 'tOU 7tEtpaµa'tO� <j>atVE'tClt <r'tOV napaKU'tffi 7ttVUKU omA.ij� EtcrOOOU:

1 2 3 4

1,1 2, 1 3,1 4, 1

2 1 ,2 2,2 3,2 4,2

3 1 ,3 2,3 3,3 4,3

4 1 ,4 2,4 3,4 4,4

n = {(1 , 1 ), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4) } Km N(Q) = 16

E>a 7tpE7tEt (1 '1 pi'JITJ +2'1 pi'JITJ)�4, 07tO'tE: A = { ( 1 , 1), ( 1 , 2), ( 1 , 3), (2, 1), (2, 2), (3, 1 ) } Km N(A) = 6 y) E>a 7tpE7tEt l '1 pi'JflJ=2'1 pi'JITJ, 07tO'tE: B = { ( l , 1), (2, 2), (3, 3), (4, 4)} Kat N(B ) = 4. o)

fta 'ta evoexoµeva r, � KClt H txouµE 'ta E�ij�: .1 r = A' = { ( 1 ,4), (2,3), (2, 4), (3, 2), (3, 3), (3, 4), (4, 1 ), (4, 2), (4, 3), (4 , 4) } Kat N(A') = 1 0. 2. E = AuB = {( 1 , 1), ( 1, 2), ( 1, 3), (2, 1), (2, 2), (3, 1), (3, 3), (4, 4)} Km 1'(E) = 8 3 . H = A n B = { ( 1 , 1 ), (2, 2)} Km N (H) = 2

EYKAEIAHI: B ' 104 T.4/44

Ma&qµaTlKU yiu 'ft)V r ' At>Ktiou

_\ O'KTJ O'TJ 51 o 01:1yµanK6.;

xropo-.; n nt1paµaTo� Tl)x11� µt not1 O"To1xdrov nH180.; 7tt7ttpaaµb·o ntpU..O µ p«vo Ta MM 1:voqoµ£Va ro., ro2, 0>3, 0>4, ros Kat ro6 µ1: n18av0Tl)T£�: P( 0>1)=0,22,

P(ro2)=0,1 4, P(ro4)=0,l6, P(ros)=0,25, P(ro6)=0,15 . 'EO"Tro T« m>v8na evoqoµtv« A={ro ., 0>4}, B={ro2, ros}, r={roi,0>3,0>6} Kat A={ro2, 0>3, ros, ro6} . Na ppe8oi>v 01 m8avoT1)T£� Trov tvoqoµtvrov A, B, r, A. Aua11

Ano tov a�troµanKo optcrµo yia ta EV&txoµEVa ClUta icrxt'.m :P( ro , )+P(ro2)+P(ro3 )+P(ro4)+P(ros )+P(ro6)= 1 , apa P(ro3 )=0,08. Enimi<;: P(A)=P(ro1)+P(ro,22+0, 1 6=0,38 P(B)=P(ro2)+P(ro5 )=0, 4+ 0,25=0,39 P(f)=P(ro1)+P(ro3)+P(ro6)=0,22+0,08+0, 5=0,45 P(L\)=P(ro2)+P(ro3 )+P(ro s )+P(ro6)=0,62 •

•

A O' K'l<Jll 611

Av A, B evoexoµeva evo� 01:1yµaT1Koi> xropov n � P ( A u B ) = , P( B ') = _! , P( A (I B) = _!_ , 4 3 4

Kat

vu pp1:8oi>v Ot 1tl8UVOT1)T£�: a) P(B ) p) P(A) y) P (A - B ) o)

t) P(A '(I B ' )

<JT)

P(B - A) P [(A - B ) u (B - A) )

A6<J'l

«) fta avti.0ttCl EVbEXOµEVCl lCJxQEl on:

P(B) = 1 - P(B ') tnoµtvro<; P(B) = 1 - � = _!_

3 3 Ano tov npocr0tttKO voµo tcrxi)tt on:

P(A u B)

&riA..

P (A) +

P (A) + P (B) - P ( A n B)

� - : � P (A) = ! + : -�

TtA.tKa P (A) = � 3 y) EniITT) <; tcrxi)tt on:

P(A - B)

A ElVCll acruµ�i�acrm, apa P[ (A - B) u (B - A)] = P( A - B) + P(B - A) = 5 1 6 1 ' + - = - = - A.oyro trov y) ' &).

O'T )

Ta A - B KCll B

12 12 12 2

A<JK'IO'll 711

AivoVTat TU £VO£XOJ1EV« 0£l"(JlUTlKOi> xropotl

P{ A U B ) = 0,4 , P{A n B ) = 0,1 5 , P(A') = 0,75 , P(r U �) = 0,8 P(r n A ') = O,S ,

�i�� +

1tl8UVOT1)T£� TO>V tvoqoµtvrov

P ( A) - P (A n B) 1 5 2 L\riA.a&i) P(A - B) = 3 4 U o) Oµoiro<;: P(B - A) = P (B) - P ( A n B) 1 1 1 L\riA.a&i) P(B - A) = 3 4 12 t) P( A 'n B ') = P((A u B) '] = 1 - P(A u B) 3 1 L\riA«&i) P(A ' n B ') = 1 - = 4 4 =

••

vu ppdloi>v

A, B, r Kat A.

AU <J TJ •

•

fta ta cruµnA.riproµanKa EV&txoµEVa A Km A' tcrxi)tt: P( A) = 1 - P( A ') = 1 - 0 , 75 = 0 , 25 rm Mo EV&cxoµEVa A KCll B EVO<; &ttyµattKOU xropou Q . 'Exouµt:

P(A U B) = P(A) + P(B) - P(A n B) � P(B) = P(A U B) + P(A n B) - P(A) � P(B) = 0 , 4 + 0, 1 5 - 0, 25 = 0 , 3 rm Mo EV&cxoµEVa r KCll L\ EVO<; &ttyµattKOU

•

xropou n, exouµc: P(r n � ·) = P(r - �) = P ( r) - P ( r n �) � P(r n �) = P(r) - P(r n � ·) = P(r) - o, 5 (1) (I)

EniITT) <; P(r u L\ ) = P(r ) + P ( L\) - P(r 11 L\) � 0, 8 = P(r) + P( L\ ) - P( r ) + 0, 5 � P( L\) = 0, 3 . , 5 AKoµri ' P(r) = - � P(L\) 7 --

P(r) = _?_ P( L\ ) = _?_ 0 3 = _!2_ = ]_ 7 7 7 14 ·

A, B, r Kat A £VO� n. Av urxi>ot1v:

A<JK'l<Jll g11

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, , po-,r £lVat: ·� VU £lVat J11tOtlKU11.l

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EYKAEIAHI: B' 104 T.4/45

Ma9ttf1«TiKa yia T1JV B ' AuKtiou

At>mi

E>eropouµe ta. evoexoµeva. A: «to µnouK6.At eiva.t M:.uKo» µe P(A) 20 , r : «to µnouKa.11.t etvm yKpt» 'I � -

= 40

' '\

= 4010 , II : «to µnouKa.11.t etva.t npa.cnvo» µe P(II) = _!:__:_!__ Ka.t Z: «to µnouK6.'At eivm po�» µe 9/... + 2 P ( Z) = A, , 'A e 9? .Enicni� P ( IIuZ ) =.!Q_40 3'A + 4 a.<pou ta. npacnva. Ka.t po� µnouK6.'Ata. µa.�i eivm 1 0. EnetMl evoexoµeva. II, P eiva.t a.cruµpipa.cna. ea. txouµe: P(II u Z) = P (II ) + P(Z) => µe

P(f)

' '\

ta.

10 'A - 1 + 'A => ... => A = - 4 T1, 'A = 2 . ' 40 9'A + 2 3'A + 4 7 fta. A = _i txouµe: P ( ) _!._ < 0 , onote TI nµij 4 7 4 A, = a.noppintetm. fta. 2 eivm: 7 P( = 2 = _3_ (1 ) v x eivm to n'Aijeo� 3 · 2 + 4 10 trov po� µnouKa.'Atrov, tote = � (2). Ano n� 40 ( 1 ), (2) ppicrKouµe: � = _3_ , OT1'Aa.oij x 8 , onote 40 1 0 -=

A, =

A P(Z)

po� µnouK6.'Ata. eivm 8 Ka.t enoµevro� ta. npacnva. eivm 2.

ta.

f\. O'KTI O'TI 9

I:t: tva m>Vt:pyt:io l>1tclf>XOVV OXllµ«T« 1tf>O� t:1ttO'K£l)ll 1tov <it:v Af:tTovpyo'6v K«vov1Ka. Axo T« ox11µ«T« fll)Tcl TO 65% BX£l µrixavol..oytKO

1 4

1tpoPJ..ri µ«, TO -

fll)TOOV txt:t flUKTpoA.oytKO

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'Ecrtro to evoexoµevo «to O.UtOKlVTjtO va. na.poucr16.�e1 µT1xa.vo'Aoy1Ko npo P'ATlµa.» µe P( )=0, 65 . 'Ecrtro to evoexoµevo «to O.UtOKlVTjtO va. na.poucn6.�e1 T1AeKtpoA.oy1Ko npoP'AT1µa.» µe P( )=0,2 5 . Enicni�, icrxt>eiP(A n B) =0, 1 0. a) To evoexoµevo «Na. na.poucr16.�et µT1xa.voA.oy1Ko

A B

Ti TtAeKtpoA.oytKo npoP'ATlµa.» eivm to A u B onote P( A U B) = P(A) + P(B) - P(A n B) 0.80 p) To evoexoµevo «To O.UtOKlVTjtO va. µTIV na.poucn6.�e1 µT1xa.vo'Aoy1Ko ii T1AeKtpo'Aoy1Ko npo P'ATlµa.» eivm to (A u B) ' onote : P ( ( A U B ) ' ) 1 - P(A U B) = 1 - 0.80 0.20 y) To evoexoµevo <<Va. txei to omµa. µovo µta. PA.6.PT1» eivm to (A - B) U ( B - A) enoµevro�, yta. tTJV a.vticrtoixTI n19a.votrita. txouµe: =

P ( (A - B) LJ (B - A) ) P(A - B) + P(B - A) = P(A) + P(B) - 2P(A n B) = . .. 0, 70 . =

f\. O'Kll O'TI 1 0 '1

f(x) 4x3 - 5x2 + x ln K + 2017 , x e 9t,K > 0 . a) Av f '(l) = 4 , vu ot:t;nt: OTl K = e 2 • =

'EaTro

p) 'EaTro n o ot:1yµaT1Ko� xropo� £Vo� xt:ipaµaTO� rixri�. Av A, B £V<it:xoµ£Va Tov n µt: A c B Km In K = 2 vu vxol..oytaTo'6v 01 1tt9avoTflT£� P (A) , P ( 8), P (A n 8), P (A LJ B) ,

P ( A n B') ,

oxov

P(A), P(B)

01 9fot:� T01ttKrov

Tfl� f.

Aua11

Eivm: f ' ( x ) l 2x2 - I Ox + In K onote: f ' ( 1) = 4 =::} 1 2 - 1 0 - In K = 4 =::} K = e2 =

p) fta. ln1r-2,

txouµe: f ( x ) = 4x3 - 5x2 + 2x + 20 1 7 . Onote:

f '(x) = 12x2 - I Ox + 2 = 2(6x2 - 5x + 1 ) Kat .

• • •

f '(x)

�

x _!_ � x .!.. 2 3 =

( ) (� ,+oo) 0 � x e ( ± , �)

f'(x) > O � x e -oo, ± u f'(x) <

A pa.,

'r na.poucna.-,e1 tomKo µeyicrto yta. x = -1 3 ,

Kat tomKo eA.axicrto yia. x _!_ . Enetoij ea. eivat

P (A) � P (B) .

cB,

Onote P(A) = _!_ Ka.t 3

P(B) = _!_ Enicni� A c B => (A n B) = A Kat 2

EYKAEIAHI:. B ' 1 04 T.4/46

Ma011JUlTtK0: yia 'TtlV r AllKtlOtl

1 P(A n B) = P(A) = - , 3

'Etcn

(A u B) = B .

P(A u B) = P(B) = _!_. T€A.oi; 2 P ( B n A ' ) = P ( B ) - P ( B n A ) = _!_ _!_ = .!.. . 2 3 6 _

AGKT)Cfl) 1 1 1]

r..a Ta "EV0£XOJ1EVa B EVO<; 0£l"(JlaTlKOV xropov iaxvovv: P(A) = o, 76 Km P(B ' ) = o,65 . a) Na .:;tTaatTt av Ta B dvm aCf'l>µpipaaTa. p) Na mrood;tTt OTl: P(A u B) � 0, 76 . y) Na anood;£Tt OTt: 0, 1 1 ::;; P(A f1 B) ::;; 0,35 . o) Na anood;tT£ OTl: 0,41 ::;; P(A - B) ::;; 0, 65 .

Aua11 a) 'Ecmo

A, B

6tt ta eivm acruµpipacrta t6te A n B = 0 o1t6te P(A u B ) = P(A) + P(B) = 0, 7 6 + 1 - P(B ') = 0, 7 6 + 1 - 0, 65 = 1, 1 1 ato1to, apa ta oev eivm acruµpipacrta.

A, B

p) foxi)et:

A c (A u B ) => P(A) ::;; P(A u B) => P(A u B) � 0, 76 .

y) Eivm P(B) = 1 - P(B ') = 1 - 0, 65 = 0, 35 . E7ttcrl)i;: (Ar1 B) s;; B => P r1B :::; P(B) => P r1B :::; P(A u B) $ l => P(A) + P(B) - P(A n B) $ l => 0, 76 + 0, 35 - 1 :::; P(A n B) => 0, 1 1 :::; P(A n B apa teAtKa 0a tcrxi)et: 0, 1 1 $ P(A n B) $ 0, 35 .

(A )

o) lcrxi)et:

(A ) 0,35

(I)

AGKT)Cfl) 1 2'1

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Aua11

a) 'Exouµe:

P(B)::;; P(Ar1B)=> 1-P(B)sP(AnB).

(l )

Dµroi; P(A ') $ 0, 2 => 1 - P(A) $ 0, 2 <::::> P(A) � 0, 8 Km Myro tou a) P(B) � 0, 5 Km µe 7tp6cr0ecrl) Kata µ€A.ri: P(A) + P(B ) � l, 3 o1t6te Myro trii; 1taipvouµe P( A u B) � 1, 1tOU eivm ato1to o1t6te ta oev eivm acruµpipacrta.

(1)

3>1 A,B U1t60ecrl) exouµe P(A ') $ 0, 2 (2) Km

y) A1t6 tTJV ' P(B) � 0, 5 => ' tou a) etvm Myro

1 1 $ - (3). P(B) 0, 5 Me 7toUa7tA.acnacrµ6 trov Kata µeA.ri , P A ) 0, 2 P(A ') 2 7tmpvouµe: ( $ - => $- . P(B) 0, 5 P(B) 5 --

(2),(3)

AGKT)Cfl) 13'1 a) 'EaTro 11 avvapTT)GlJ

f(x) = x2 - x3 , x e [o ,1]

Na J1EAETT)8d ro<; npo<; TT) µovoTovia Km Ta aKpOT«Ta. EVO<; 0£l"(Jl«TlKOV p) nu Ka8t EVO£XOJ1EVO

xropov n vu otLX8d oTL: P 2 (A) · P( A ') ::;; AUGT) a) H

o7t6te, •

("'\

P(A-B) = P(A)-P(A B) = 0, 76 -P(A B) (1) . A6yro tou y) eivm: 0,11::;; P(A r1B):::; 0,3 5 =>-0,35 $-P(Ar1B) $-0,11 => 0, 76-0,35 s 0, 76-P(A r1B):::; 0, 76-0,11 => 0,41s0, 76-P(A n B):::; 0,65 =>0,41$P(A-B)s0,65 .

s;;

An B B => P(A n B)::;; P(B) o1t6te 1-P(B)::;; P(B) => P(B);;::: 0,5 'Ecrtro 6tt ta A, B eivm acruµpipacrta t6te A n B = 0 o1t6te P(A u B) = P(A) + P(B)

Dµroi; 1taipvouµe:

• • A

•

f '(x) = 2x -3x2 f'(x)=0<:::> 2x-3x 2 =O <:::> x =O i] x = �3 f '(x) 0 0 x 23 f '( x) 0 -23 x $ 1 2

f eivm 1tapayroyicnµri µe

n.pa ri f crto '

�

<=>

-3

' = 7tmpvet tTJ µeytcrtri nµri trii;.

x 0

x E

�riA.aoi] yta Ka0e

[ 0, 1 ] 0a tcrxi)et

f(x) s r (%)

f ( x ) s _i_27 . Av 0foouµe P(A)=x t6te P(A') =l-x µe 0 $ $ 1 , t6te apKei va Oe�ouµe 6tt: x 1 -x) ::;; 247 ' x2 -x $ 247 ' f ( x ) s 274

oriA.aofi p)

(

1tou tcrx6et a7t6 to eprotrJµa a).

EYKAEIAm B ' 104 T.4/47

MaOflflU'tlKQ y1a TflV r AuKtiou

e,..avaAr\1rTI KE� AaKr\0£1 � - AvaAuan A<JK'l<Jll

a, p

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f(x) x l x - a l l x - 13 1 +

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•

f(P)= PI P - aJ = P(P-a) . 'Etm exouµe: f ' (P )=f;( P)= lim f(x)-f(p) = x-p = lim x(x-a)+(x-x-pP)-p(j3-a) (yianx> ,_. > a) = = II.m (x-p)(xx +P- P -a+l) Im (x+,_.-a+l) = =2P -a + 1 Dµma ppicrKoµm on: f' = (p) = . . = 2J3 -a -1 . 'Etot exouµ£: 2P -a + 1 =2P -a-1 Km 6.pa 1 -1 , 6.t01tO. 'Eotoo on a > p. Tot£, oµota cp06.vouµ£ 0£ 6.Tono. :Euv£m:0�: a = p, onot£ : f(x) = x l x -al+ I x -al = (x + l) l x -al . f(x)-f(a) = 'Etoi, exouµe: f '( a)=f;(a)= lim x-a x -a (x+l) - I'm x-al l Im (x+l)(x-a) =a+ 1 . x-a Dµoia: f '(a) = (a) = -a - :Euv£m:0�: a + 1 = -a -1 Km 6.pa: a = -1. Ilot£, av crovO.pTIJCfl'l f £ivm napayooyicrtµTI oto , tot£ : a = p = -1. 2 ) AVTl<JTpoq>ro�. 'Eotoo on: a= p = -1. Tot£: 2 ,6.v x�-1 f(x)=xl x +�+ l x +ll =(x +l)l x +ll = {(x+l) -(x + 1)2 , , x<-1 'EoToo on x > -1. Tot£: f (x) = (x + ) 2 Km 6.pa f'(x) = 2 ( x + I) . 'Eotoo on: x < -1. Tot£ f(x)=-(x+l) 2 Km 6.pa f'(x) = -2(x + 1). E�£T6.�ouµ£ TIJV napayooyo oTo -1. 'Exouµ£: (x +l)2 = lim (x +l)=O. li m f:(-l)= lim f(x)-f(-1) x+l x+l Dµma: C(-1)= ... =0. :Euv£1tcO�, TI crovO.pTIJCfl'l f dvm napayooyicrtµTI Km Km 6.pa £ivm napayooyicrtµTI OTO x -+ P.

x�

x -+P+

(J3)

•

x -+a+

x -+a•

x -+a+

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x -+-1+

oto -1

x-+- 1+

X-+- 1+

lR .

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a = -1 p = -1 .

6'60 <popt� 1tapayroyiatf.l'I OTO lR Km 'I avvup'T'l«Yfl dvm avvtx1}� OTO lR . Na a1to6d;tT£ OTt 'I avvupT'IOfl dvm upTta av Kat f.lOVO av 11 (J'l)VUP'T'IO'I : = dvm upTta. A\J<Jfl. 1) 'Ecrtoo 6n TI civm 6.pna. T6t£, yt.a. K6.0£ e lR Ex,ouµe f(-x) = f(x) Kat cruv�:

f g(x) 2f(x) - xf'(x)

f x -f'(-x) = f'(x) Kat f'(-x) - (-x)' = f'(x) , cl.pa TI f' civm 1t£PtniJ. 'Etm Ex,ouµE, Ka.0£ x : = = 2f x) x) = 2f -x) x) xf' (-x) (x). + x f '(- ( ( g ( g Apa, TI g civat 6.pna. 2) VTl<JTpocpro�. 'EoTOO on TI g £ivm 6.pna. E>a a1t006�ouµe on Km TI f £ivm 6.pna. 'Exouµ£ yia Ka0£ x e g(-x) = g(x) � 2f(-x) + xf'(-x) = 2f(x)-xf'(x) � x [ f'(x) + f' (-x)] - 2 [ f(x) - f(-x) ] = O (1). TIJV Tffipa crovO.pTIJcni: E>Eoopouµe q>(X) = f(x)-f(-X) (2). 'facrt a1tO TIJV (1 ), EXOUµ£ yia K6.0£ x e : xcp' ( x) -2cp( x) = 0 . Ano autiJ exouµ£ , yta Ka0£ X2cp'(x)-2xcp(x) = 0 � ' cp(x) 0 � ( cp( x) )' 0 x2 cp'(x)-(x-2 )-----4 - x x cp( ) l:uµn£p6.vouµ£ on: : =A., yia Ka0£ x e (0, +oo) x cp( ) Km : = µ , yta Ka0£ X (-oo,0), 01tOU µ, A, JR . x {A.xµx22 ,av,av x<Ox>O . :EUV£1tro�: 'Etcrt exouµe: cp(x) = { 2A., av x>O , {2µx, 2A.x, av x>O <p ( X) = Km <p " ( X) = 2µ, av x<O av x<O E�6Mou, rnm()ft TI cp" £ivm cruv£Xll� mo ( yia.Ti;) Ex,ouµe: limcp"(x) = limcp"(x) , OTIAMTt 21..=2µ Kat cl.pa A. = µ. A6yoo auTou Km 67t6t0ft cp(O)=O, Ex,ouµe: 2 cp{x)=AX , yt.a. K6.0£x e!R . Ano 'tTjV (2) Ex,ouµeyia K6.0£ x e !R : cp(-x) = f(-x) - f(x) = -<p( x) �cp(-x)=-q>(x) �A.x 2 =-A.x 2 � 2A.x 2 =0 . :Euvrnro� A,(=µ)=O . 'ETm, yia. K6.0£ x e , Ex,ouµe : cp(x)=O Kat �:f{x)=f{-x). Apa TI f civm 6.pna. &JA.a&i

E lR

yt.a.

lR :

X E JR* : _

X-+ o

EYKAEIAH:E B ' 104 T.4/48

crovrnffi

X-+ o

Ma011paTtKa yta 't'IV r' At>Kdot>

A<JKTJ <nJ 3 . Ai>o auvapn}atK; Kat g tivm optaptvt� Kat 7tapayoryiatpt� <J'rO lR pt Km yia Kci9t e lR iaxi>ovv:

f f(O)= l f 2 (x) - g2 (x) = 1

x (1) Km f(x) = g' (x) (2).

l) Na a7tooti�tTt oTt: g( x) = f' ( x) , yta Kci9t

IR .

q>=f+g Km h=f-g

2 ) Na pptiTt Tt)� avcipTt)<Jt)�: K«l fl&Tcl 1'K; K«l A tl <JTJ. I )

( 1 ), na.pa.ycoyisovta.�, txouµc ato 2f(x)f'( x)-2g(x)g'(x) = 0 ( 2) f(x)f'(x)- g(x)f(x) =0 => f(x) [f'(x)-g(x)] = 0(3) 'Eatco Ott yta eva.v a.pt0µo t<JxUEt f(a.)=0. Ano tllV ( 1) µE x=a �ptaKOµE Ka.t apa. g 2 = -1 . 'Etcrt ano ' atono. A.pa: f(x)-:;:. 0, yta KCt0E x tllV (3) txouµc yia Ka0E x e f'(x)- g(x) = 0 Km cruvEmo�: g (x) = f ' ( x) (4). 2) 'ExouµE ato R : cp'(x) = f'(x) + g'(x) g(x) + f(x) = cp(x) => cp ' ( x) = cp(x) cp ' ( x)e-x -e-xcp(x) = 0 => ( cp(x)e-x )' =0 =>cp(x)e-x =c (ce!R) Ano tllV (1) µE x=O �piaKouµE f 2 (0) - g 2 (0) 1 Ka.t E1tEt8i) f(O) = 1, EnE't'a.t ott g(O) = O . 'E't'at txouµE: cp(O) = f (0) + g(O) = 1. E�aUou: cp(O) =c. LUVE1tCO�: c=l Ka.t apa, a.no tllV : <p(x) =ex (5). h'(x) = f'(x) - g'(x) = g(x) - f(x) = -h(x) + 0 h'(x) = -h(x) Ano tllV

lR :

E lR

•

(a)

E lR

lR :

=:> cp(x) = c · e x .

(

=> h (x)e x

)

:::::>

h ' (x)e x

e xh (x) =

=0. 'Exouµc: Ka.t h(O) = f( 0) - g(O) = 1 Ka.t apa. c' 1 . LUVEmo�: h (x) = e- x (6). Ano tt� (5) Km (6), txouµE aw !R :

h(O) = c'

f(x) = <p(x) +2 h(x) = ex +2e-x Ka.t g(x) = cp(x)-2 f(x) = ex -2e-x

;\<JK11 m1 4 .

y + 1 ,ye!R. f(x)=2vx + -;-1 - 2,xe(O, +oo) Ka.t g(y)=7 'Ernt a.no (2), txouµE: f(x)=g(y). (3) - H OUVCtPtllITTJ f EiVa.t na.pa.ycoyimµri CHO ( 0, 0, av. x> 1 ..., 1 1 1 xµE: f ' ( x ) = 2 2vx - -x 2 = x2 = 0,0, 6.va.v' Ox=<x1< l r

t11

Na pptiTt TO'\l� api9µoi>�

y e JR , tTa• roaTt va iaxi>ti: (2x.Jx + 1 - 2x)eY = (y + l )x .

f f(l )=xe(O, 1. oo), +

:EuµnEpa.ivouµE ott rt cruvaptll ITTJ EXEt EA.axtatll ttµi) aw iITTJ µc :EuvEmo�: yta Ka0E µE t0 µovo yia H cruvaptll art Eivm napa.ycoyicrtµT] aw lR µt: a.v �. a.v y=O < a.v •

f(x)�lx= 1,, = x=l. g 0, y<O Y Y -(�: e l)e g'(y) = e = - e = 0, 0, y>O

0 g

-x

g g(O)= 1. ye

:EuµnEpa.ivouµt ott rt cruvaptllITTJ EXEt µEytatll ttµi) cno iITTJ µE :EuvEmo�: yta Ka0E lR , µt w=µovo yta. 'ExouµE A.otn6v A.Oyco Ka.t tll � (3): f (x) = 1 x=1 :::::> 1 ::; f(x) = g(y) ::; 1 :::::> y=O g(y) = 1 Avtta't'pocpco�. Dncos �piaKouv Ka.t . EUKoA.a. µE TJ taxUEt. A.pa., ot sTJtOUµEVTJ a.pt0µoi Eivm: KClt •

y=O, g(y):::; 1,

{

y=O ( 1)

{

Y"".0.

x= 1 x= 1 y=O.

A<J KTJ O"TJ 5. Na PptiTt Tt)� 7tapay0>yiaipt� auvapn}atK; : IR --+ IR ' tT<Jl ro<J'rt "(l« Kci9t E IR va l<JJ(i>tt:

f x ( x2 +1)[r(x)-ln�x2 +1 J= x[x-( x2 + l)f'(x)J (1)

f ( 1) ( x2 + l) f(x) + x ( x2 + l) f'(x) = ( x2 + t) In �x2 + 1 + x2 x2 :::> f(x)+xf'(x)=ln .._ix- + 1 + -x2+1 => (x)'f(x)+xf'(x)=(x)' ln �x2 + 1 +x ( ln �x2 + 1 )'

'Ea't'co ott µta. cruvaptll ITTJ nA.11poi 't'tS OEOOµEvEs cruv0i)KEs. Ano tllV txouµE yta KCt0E xeR: Au<JlJ .

x ( O,+oo) e

(1)

x>O y e ( 1) 2� + _!_ - 2= ye+lY . (2)

'Ea't'co ott yta Mo apt0µou� Ka.t TJ taOtll't'U taxUEt, onotE �a. txouµt: A (lm1 .

+oo )

r:l

:::::>

•

E>EcopouµE tt� cruva.pTI)crnt�:

�

EYKAEIAHE B ' 104 T.4/49

:::::>

' => (x f ( x ))1 = ( x ln � ) =>

O < a

xf(x) = x ln .Jx 2 + 1 + c (c e JR.) (2). =0 C = 0. e R : xf ( x) In .J 2 x [ f(x) - ln .Jx 2 + 1 ] = 0

Ano t'lV (2) µe ' X ppicrKOUµe Etcrt, U7t0 t'lV (2) exouµe yia Ka0e x x x + 1 Km (J1)Vem:O�: (3). Ano t'lv (3) µex * O , exouµe: f(x) = ln-1x2 + 1 (4). E�aUou, e7tet0fi TJ f eivm (J1)VeXfi� crto 0, exouµe: f(O) = lim f (x) � limln � = 0 . Luµnepavouµe ott: f(x) = In .Jx2 + 1 , ta Ka0e x Dnco� ppicrKouµe eUKoA.a, TJ (J1)VUPtTJOlJ nou ppi]Kaµe: f(x) = ln .Jx2 +1 nA.ripoi tt� OeOoµeve� (J1)V0i]Ke� Km apa eivm TJ µovaotKi] sT}touµevri. =

X->0

•

R .

A (j KTJOlJ 6. Mia Gl>VUPTTJ<rll tivm opt.a�, napayroyiat.pl) Km KVp'Tfa mo lR . l:To 0 F:x,ti T07t1.KO dciXt.<JTO, 1) dvm Gl>V£Xfa� <JTO O · Km yt.a Ka9t x E lR t.<JXl}ti: > 0 . Na a7tOOti;£Tt O'Ti: 1 ) H Gl>VUPT1J<rll mo 0 f:/.tt. oA.t.KO tMX,1.aTO. 2) H Gl>VUPT1)<J11 : dvm KVp'Tfa mo lR 1 .\ lJ(j 'l · ) f 1R f fI ! R .

ff(x)'

g(x) = f2 (x)

Enetoi] TJ (J1)VUPtTJOlJ eivm KUptfi crto ' E1tetm Ott: E7tet0fi TJ ex.et to7tlKO elaxicrto crto 0, faetm on: f'(O) = 0 . 'Etcrt exouµe: x < 0 => f'(x) < f'(O) = O ::::> f'(x) < 0 . x > 0 => f'(x) > f'(O) = O ::::> f'(x) > 0 . Apa: f J ( 0] Km f ![ 0, +oo) . Luµnepaivouµe Ott TJ (J1)VUP't'lalJ f O"tO 0 exel OAlKO eAUXtcrtO. -oo ,

�1 -x .L � -x �

g eivm optcrµ� Km (J1)VUP't'lalJ 7tUpaycoyicrtµTJ O"tO R ' µe: g'(x) = 2f(x)f'(x) . ea anooei�ouµe ott: g' ! R . E>ecopouµe ouo apt0µou� a,p R . 'Exouµe: 2)

a

> f( P) {f(a) f'(a) < f'(P) < f'(O) = O

{-f'(a) f(a) > f(p)(> 0) => > -f'(p)(> 0)

=> f(a)f'(a) < f(P)f'(P) => g '(a) < g '(P) . g'!(O,+oo) . ' TI g ! ( -oo, 0) g' 0, TI g' ! R g R.

Luµm:paivouµe ott: Km e7tetofi Km (J1)VUPtTJOlJ eivm (J1)VeXfi� crto faetm ott: .Apa (J1)VUPtTJOlJ eivm KUptfi crto A (j K'l (j'l 7. Na l>7toA.oyiatTt TO oJ.orl.fapropa:

n'2 aT) µx + Pcruvx I= J dx , o7tov 'Y1l µx + ocruvx 0

yo>O. A lJ(j 'l ·

13, .y, o e R

(J1)VUP't'lalJ f ( x) = rri µx + ocruvx Otatllpei crta0epo to npoariµo t'l� crto [0, ;] , a<pou yo>O Km 11 µx 0, cruvx 0 , xcopi� va icrxt)ouv Km ot ouo tcrO't'lte� (J1)yx.povco�. Apa TJ + pcruvx eivm (J'\)Vexil� crto [o, 7t] h ( x) = mi µx yT}µx + ocruvx 2 E�etasouµe av unapxouv npayµattKoi. api0µoi A. , , Km µ tetoiot cocrte, yia Ka, ee x [o , 2"n ] , va l<JXUet:, ariµx+p(J'\)vx=A.(rri µx+O(J1)VX)+µ(yriµx+O(J1)VX) ' (1) Ilpo� touto, apKei yia Ka0e x [0, ;J , va icrxt'>si: ariµx+p(J1)vx = (A.y-µo)riµx (A.O+µy)(J1)vx, apKei: {pa=='Ao"Ay +- µyµo ' apKet.'· ( 'I -_ ayy2 ++ opo ' µ_ pyy -+ aoo ) 'Etcri, A.Oyco Km t'l� (1),exouµe: l = AT dx + µT (YTJ µx + OO'uvx)'dX = YTJ µx + "-2 + µ[lnlYTJ µx + OO'uvxl]� = = ayy2 ++ 0po2 . 2:2 + py12 -aO + 02 A7tepv t7ta=-o, tote exouµe t'lV(YTJµU7tx+O(J1)VX) AoUcrtepTJ' ' p=y'ariµx+p(J1) t V X = COOl) npoK\Jntet Oe Km ano t'lV yeviKfi2 7tepimcoari, mpou, tote, . 'A = -0y12 ++oy2o = 0, µ- 1y2 ++ o022 = I H

�

OOuVX

. 1nr

a·

•

A (j K 'l (j 'l 8. Na l>7tO#.oyiatTt TO oJ.orl.fapropa: I

=> -f( a)f'(a) > -f( P)f'(P) => g '( a) < g '(P) . g' !(-oo, 0)

LUµ7tepaivouµe Ott:

{(O(O <)f(a) < f(p) = f'(O) <)f'(a) < f'(p)

Ma9ru1anKO. y1a 'TflV r A1lK£lo1)

.Eniari� exouµe: EYKAEIAH� B '

A \J(j11.

l = J �x l+x 0

h t'lV (J1)VUP't'lalJ nou eivm crto oA.oKA.fiOvoµasoµe pcoµa. Ilapat'lpouµe ott: 104 T.4/50

Ma911pa1'tKa yta 'tllV r ' AllKtioll

I = J 2(1 +1 x 2 ) (1 + x 2 )' dx . 0

Eivm cpaw:p6 6n, 0Ecoprovtm; ni:; cruvaptiicmi:;:

.J3

I = J h (x)dx = J f (g(x) )g '(x)dx

Apa:

1 · tx = . 1 ( f(x)dx = �x = x - 1 + -Km f(x) = , tcrxUEt: = g(O) X + 1r X+1 I I 2x 2 h(x) = f (g(x) ) g'(x) ym Ka0E x e L'l . EmnA.tov x2 1 3 = - - x + ln ( x + l) = ... = - + ln2 2 icrxuouv 6Af:i:; ot uno0focti:; tou 0Ecopilµatoi:; 2 I avnKatacrtacrri i:;, 8T1A.a8il TI g Eivm optcrµtvTJ Km ACJKTJCJTJ 1 0. Na t1nol.oyiat1'E Ta oJ.oKJ.riproµa1'a: napaycoyicrtµTI crto L'l, TI g'(x) =2x, optcrµtvTJ Km 4 l cruvcxili:; crto L'l Km TI f Eivm optcrµtvr, Km cruvcxili:; I J£ dx Kat L = J .Je• + ldx X+1 crto g(L'l)= [ l , 2 ]. ea E<papµ6crouµc, AOt1tOV, to g(x) = l + x2 l [O,l] = Ll

[

0EffipTlµa avnKatacrmcrri i:;, 0Ecoprovtai:; to 8ocrµevo oA.otlilpcoµa coi:; to nproto µeA.oi:; tou 0Ecopilµatoi:; autou. 'ExouµE:

I g( I ) I 2 1 = I = Jh(x)dx = Jf ( g(x))g'(x)dx = J f(x)dx = J� 2X

g(O)

= .!.[lnx)2I = _!_ ln2 = In J2 2 2

B 'tporroi; (Xropii; TO 6 Eropri µa av1'tKa1'a<J1'amii;) 'ExouµE: on6tE

[

(1 + X ) 1 = -ln(l + x2 ) , h(x) = 1 2 l + x2 2 l

[�

l = 1n(l + x2 ) = ln J2

ACJKll CJTI 9 .

Na t1nol.oyiat1'E 1'0 ol.oKl.1lproµa: .J3

x +1 x� I = J ...; - 1 dx . 1 + .Jx 2 + 1

:EKonEuouµE va E<papµ6crouµE to 0EffipTlµa avnKatacnacrri i:; Km µaA.tcna va 0Ecopil<muµE to 8ocrµevo oA.otlilpcoµa coi:; to nproto µEA.oi:; tou 0Ecopilµatoi:; <lUtOU. OvoµasouµE h tTJV cruvaptTIITTJ nou Eivm crto oA.otlilpcoµa. ea npfaEt va ppouµE Mo cruvaptiicrcti:; g I 1, = a = l, p = Km f, ot onoiEi:; va 1tATlpouv ni:; uno0focti:; tou 0Ecopilµatoi:; autou Km EmnA.tov va tcrx;\>Et: h(x) = f(g(x) )g'(x) , ym Ka0E x e L'l . :EtT1V nEpintcocn1 µai:; ev8EiKVUtm, npoKEtµevou va anaUayouµE an6 ta ptstKa, va 0EcopilcmuµE tTJ AUCJTJ .

[ .J3] L1(

cruv<iptTJITTJ

�x) �, on6tE g'(x)

cruvcxili:; crto L'l Km

.J3 )

�

=� 2 x2 +1 vx2 +1

h( x) = x.Jx2 + 1 .Jx2 + 1 = ( .Jx2 + 1 f = f(g(x)) = g ' (x) 1 + .Jx2 + 1 x 1 + .Jx2 + 1 = gz (x) ) 8T1A.a8T1, f ( x) = x2 , nou Eivm l + g(x l+x cruvcxili:; crto g ( L'l) = [ l, 2] . ·

ln8

ln3

:EKonEl>ouµE va E<papµ6crouµE to 0EffipT1µa avnKatacrtacrri i:; Km µaA.tcrta va 0Ecopilcrouµc to 8ocrµEvo oA.otlilpcoµa coi:; to OEUtEpo µEA.oi:; tou 0Ecopilµatoi:; autou. 'Exouµc:

AUCJTJ .

1 . H cruvaptTJITTJ f( x ) = --CautTJ Etvm optcrµEVTJ , vx + 1 Km cruvcxili:; crto OtUcrtlJµa [ 0, 4] . ea 1tpEnEl va ppouµE µta cruv<iptTJITTJ g optcrµtvTJ crE eva KAf:tcrt6 OtUcrtlJµa µE aKpa a,p tEtOta cOCJtE g(a)=2 , g( p )=3 Km TI onoia µasi µc tTJV f va 1tATlpouv ni:; uno0focti:; ,

tou 0Ecopilµatoi:; avnKatacrtacrri i:;. fta va a1tA01tOtTJ0Ei to p�tKO 0Ecopouµc tTJV cruvaptlJITTJ : g(x) = x2 I [ 0, 2 ) . Dncoi:; ppicrKouµE EUKoA.a 1tATIPOUVtm 6Af:i:; ot uno0foEti:; tOU 0Ecopilµatoi:;.

I= g(J2) f(x)dx = J20 f (g(x) )g '(x)dx 2( 2 x 1 } = =2J 1-- x = =Jo2 N1 + l ·2xdx = 2J�x x+l 0x+l 0

'Exouµc A.otn6v:

]

g(O)

2[ x - ln ( x + l )J: = ... = 4 - ln9

2) Dncoi:; 1tpOTJYOUµevcoi:; ea E<papµ6crouµE to 0EffipTlµa avnKatacrtacrri i:; 0Ecoprovtai:; Km n<iAt to oocrµevo oA.otlilpcova coi:; to OEUtEpO µEA.oi:; tOU 0Ecopilµatoi:; autou. 'ExouµE: f(x) =Jex + 1 1 [1n3,ln8], optcrµtvr, Km cruvcxili:;. Tropa, ym va anA.onotT10Ei to ptstKO, 0EcopouµE tTJV cruvaptlJITTJ : g( x) = ln( x 2 - 1) I [ 2, 3) , TI onoia 6ncoi:; ppi<JKouµc Ei>KoA.a µasi µE tTJV f nA.T1pouv tti:; uno0foEti:; tou 0Ecopilµatoi:;. 'ExouµE A.om6v:

J f ( x ) d x = J2 f ( g ( x ) )g '( x ) dx g(2) x2 ( 1 + -x2-1- 1 x = 2x dx=2f-T-dx=2f = fvelr(• -ll +l· 1 x x-1 J 2 2 l } [ ]� f ( 1- - x= 1 x + l x [ In ( I ) In ( x l ) ]� = . . = 2 + f g(J)

EYKAEIAHI:. B ' 1 04 T.4/5 1

x -

Ma9qpa'ttK0: yia 'tflV r AtlKtiotl

A� 1oaru.Jc iwTt� EK0tT1 Kt� Kai /\oyap 1 0 1-11 Kt� av1a6Tt'\Tt� Ilanao11 µ11Tpiot> BaaiM:ioc; - Avi:monot>l..o c; KrovaTav-rivoc; A<JKTt<Hl 1

IJ, x E (0, +oo)

A u a11 :

Km a * f3 , TOTE 1.CJXVEi:

(;::r >(;J.

0eropouµt

f: [O, +oo)

( x) = [

napayroyo f'

IR,

f(x) a ( l}+x) tn( +x ) j}+x e µe

(a+x) l3 +x

niv

m.>Vapnicr11

I Hx

KCll

= (:::r {( P + x) ' m(:::) + ( � + x) [m(:::)J}

f(x , ) > f(x2 ) � f(x2 )-f(x 1 ) < 0 � f(x , ) < 1 � f(x 1 ) < f(x 2 ), 1tpayµa � f(x 2) KCll

e f( x , ) f( x , l

<1� at01tO.

Apa f(x1) < f(x 2 ), 011A.aofl 11 O"UvapniO'TJ €tVCll 'YVT)O't(J)� au�O'UO'Cl O'tO [0, +oo ) B'

Tpo7toc;

fta g

g( x) =

O"UvapniO'TJ

! [ 0, +oo ) = �'

txouµe

( X ) ex ( X + 1 ) =

= =( :: : r H:: : ) + �::} ym Ka9e x . E�aUou ·· x 1 = g ( f(x , ) ) , x 2 = g ( f(x 2 ) ) , on6te: fta Ka9e t>O txouµe: 0:5:x1 < Xz �g( f(x, )) < g( f( Xz )) �f( x, ) < f( Xz ) . � 1 1 1 , t:5:--l�J t�l n µt lO'OV Jn-:5:--l�-J n --, t t t t yvrocrt6v ! � , t6te Ka9e y" y 2 Ll µ6vo ym t= 1 , µe t = -a+ x -:;:. 1, onote a+ x tcrx\>et: y, < y2 �g( y, ) < g( y2 ) avrurr a -:;:. � -fta Ka9e x 0, txouµe: P+x P+x -x- � a+ x ) >l - p + x � 1n a+ x ) > a- P � = >O f{x)ef( x ) =x�f x) ( txouµt: 1n( f(x) P + x a+x ( P + x a+x x ) + P -a >0 � In ef(x) = In r(x) � f(x)=lnx-lnf(x)� � ln ( a+ p + x a +. x � lnf ( x)+f ( x ) = lnx (2) �l')A.aM1 txouµe / ( x ) > ym Ka9e x ;::: 0, AUa: t > 0 � In t :5: t - 1 < t . Apa.: 01t0t€ l') <n.>VaptTIO'T) f €lVCll 'YVTJO'l(J)� ClU�O'UO'Cl crto [ 0, +oo) . Apa: lnf( x) < f( x )� lnf( x ) + f ( x) < 2f(x )�(2) lnx x > 0 => ( x) > f ( 0)=> (::: r >(; )' lnx < 2f(x) � f ( x) > T f : [O,+oo ) � JR , lnx=� on6te KCll f(x) =-+<xl foxl>et f( ) I

'tO

A 'l 'I � : .rvvw.

mpol>

�o

yia

po<p roc;.

A 1-1

Km e f

( x)

�

A<JKTt<JTt 2 : Av 11 (Jl)Vclp"t'l<J'I

1.KQV01f01.Ei T'IV CJXE<J'I f (x)e

x ;::: 0, va oi:ixOEi 6n:

x, yw.

Kcl9E

I. H (Jl)VclPT'l<Y'I f dvm 'YV'IITTO>c; av�ooaa,

I I . lim

x -++«>

f {x) = +oo

Aua11 I.

I l l . Jim

x -++oo

0, µt X1 <

f ( x, ) < f ( x2 )

'Ecrtro Xv Xz ;::: 0€t�O'Uµt on:

f {x) In X

= 1.

Xz .

ApKei va 'Exouµe: •

f(x1)e f(x i ) x1 KCll f(x 2 )e f(xz ) x 2 . ( 1 )

•

X2 > X1 � 0 .

µt Av f(x1) at01tO.

f(x2 ), tote x1

x 2 , npayµa.

EYKAEIAHI:

III.

Jim

X-++00

rm Ka9e x

l im

x-++oo f ( x ) = +oo lim (ln t ) ' lim -1 = 0 , 1-++oo ( t ) ' 1-++oo t

f(x) x In

> 1,

Jim

1 f(x) f(x) + f(x) 1 +

txouµe

Km µt

___

In f(x) • f(x)

X-+t«>

AUa,

t>O txouµe:

ln t(:)= 0, l1-++oo t

on6te im

crl>µ<powa. µe tov Kav6va de L' Apa:

l im

x-++oo

B ' 104 't.4/52

Hospital

lnf(x) = 0 � lim -f(x) = -1 = 1 j( ) X 1+0 X

x-+ + oo In

Ma011paTtKO: yta Tl}V r At>Kf:lo\) •

fevtKO'ttpa, Ka'tatdl'Youµc cr'ta ifoa cru µntpacrµma av f { x)er( xl =Ax <TIO [0,-+oo) , 6nou A. 0t'tud1 cr'ta.0tpa (ytmi;). Mia <JXETtKI\ avmpopci <JTO np01rro'6µEV0 TE'6xo� Ano TOV r1ropyo Ta<J<J07tO'\lW

Ero

Bljµa WV EvKA.ei<511 revxoc; 1 03 od 75, ava<pep81jKaµe <Je '!rap6µozo fJtµa we; µza aK6µ11 mfJavlj '!rape<:'1Y11<J1'/ · Me r11 &vrep11 avvj eVKmpia 88Aovµe va e'lrl<J11Wivovµe 6rz &v fJa v'lrljpxe Kav Kiv<5vvoc; '!rape<:1jy11<J11 <; av &v aA.A.a(aµe (o'!rmc; KaKdJc; yiveraz) TY/ µerafJA.11v].

x(v'1 + x2 + 1) + 1 + x2 - l :::; x, + 1 + x 2: 1, iJ J1 + x2 + x 2 0,

iJ J1 + x2

1tOU tcrxl>tt ( 0.1tOOtix0rtKt CT'tl'IV a.pm). fta X > 0 , apKti VO. Otl�OUµt on:

M--1

1n ( x + v'1 + x2 ) ;::: J1 + xx2 - I , iJ x + v'l + x2 2 e x

2 -t + 1 ::::> __!__ ;::: 1 - t, yta Ka0t t E lR . e' 1 Av 6µco� t< l , 'tO'tt e1 :::; -- . Acpou x>O 0a 1-t ln x A.oz'!rov txovµe lim g(x) =O J1+i -l <Jl+i+2x-l �( l+x)2 -1 - l+x-1 =� Me g ( x) = t tivm: X-+f<O X x x x x M--1 (Kav6vac; de L ' Hospital) Km a<pov x < e = lim f{ x) = +oo Ba eivaz Km on6'tt . x -++«> _ � - 1 x + l - v'l + x2 1 In ( x ) x lim g ( f { x)) = 0 , <511A.a<51j lim =O. X-H«> ( X) Apa x + l - v'I + x2 > 0, on6'tt apKti va E<:illov &v Atµe 6rz 1'/ avvapf1'/<J11 'lr.X. xr:----:; :::; x +vl+x' r:---2 , 6n 11 h{ x) = riµ.fx' eivm a6v8e<J11 g o f rmv l+x-v1+x· = =ri f { x) .fx' Kaz g( y) µy, alla ww x:::; x +�l+x2 +x2 -(1+x2) , iJ 'ttAtKa 1 :::; J1 + x2 f{ x) =J;. Kaz g(x) =riµx . I'evzKa hpe'!re va 1tOU tcrx(>tt. X-Ha:>

f f

ypa<povµe:

g ( f { x) ) = f, g ( x) = f ::::> Xlim lim f { x) = k, lim x-+k �Xo o'!rov <pvmKa f ( x) ;;; k Kovra ow x x-»<o

•

AO'KTJ O'll 3 : r1a Kcl9E 7tpayµaT\KO ap19µo x l<JX'6E1:

1 + x ln ( x + J1 + x 2 ) ;::: J1 + x 2 •

Aucn1 Ilpocpavro�

�l+x2 > N =lxl 2-x ::::> x +�l+x2 >0,

on6Tt txtt voriµa o "A.oyapt0µ6� mu. fta x = 0, tcrx(>tt co� tcrO'trt'tO.. fta apKti va oti�ouµt 6n:

x < 0,

J1 + x2 - 1 . Av cr'tl'IV In t :::; t - 1 In ( x + '1' 1 + x2 ) :::; x µt t > 0 , 0£crouµt t x + .Ji + x 2 > 0 , txouµc ln(x + .Ji + x 2 ) ::; x + .Ji + xz - i,

A<JKT)O'TJ 4 : Av O<a< p , TOTE l<JX'6Et: A U0"11 :

01tO'tt apKti v a oti�ouµt 6n

X + '1'l,.--+ x· ---:; - l :::; J1 + x2 - 1 , x

, rt

(

l + x2 - l x + '1'lr;--:i + x- - 1 $ ,.------:; x '1'1 + x 2 + 1

Ti x + � - 1 $

Q , 1 + x2 + 1

ApKti va oti�ouµt 6n

•

[ ]

()

( )

Bamcrt1lKaµc

crtrJ

yvcocrril

avtcr6nrm

6nou 'tO icrov tcrxl>st µ6vo

yta t=l . BtPA.toypa<pia I)

3) '

2 I2 + t1 2: 2 · 1 · t1 , D.

Mitrinovic,

"Elementary

inequalities",

Noordhoff ( l 964). Mitrinovic

inequalities",

)

�<.f!�. a a I}

Inx2 < x _ _!_, yta x =� > 1 'Exouµt: x a x xl x 1 x ln x2 = 2ln x = 2J-dt < J 1 + 2 dt = Jdt + Jc2dt = t I t I I I x = [ t ] x + -tl = X - 1 - �1 + 1 = X - �1

,.-----::;

e -t

'Exouµc:

and

Springer-Verlag,

Vasic Berlin,

"Analytic

New

York,

( 1 970).

G. V. Milovanovic, D . S . Mitrinovic and Th. M. Rassias,

"Topics

Polynomials:

Extremal

Problems,

Inequalities, Zeros" World Scientific Publishing Co.,

-I 1

Inc., River Edge, NJ, ( 1 994).

L Neype1t6V't'Tl�. l:. ruoT61touA.o�, E. rt.awaKoi>A.ta�,

A1tEtpocmK6� Aoyicrµ6�, l:uµµe•pia, A0i)va( 1 987)

II . E . TcraofooyA.ou, "Avmoni•e�", Eurograph A.E.,

A0i)va,

EYKAEIAHI: B' 104 T.4/53

( 1 993).

Ma9'qpaTtKa yia TTIV r · Al>Ktio\J

Tiapayouac� 1. a )

r ( x) = x

Aivt:-rm tt <n>vapTt)<Jt)

pt: x e lR .

i) Na ppt:Od tt f' . ii) Na ppt:Od

cruv

( 3x + 1)

p(a napayovaa -rqi; x = X T) µ 3x + 1 GTO JR . Aivt:Tm <n>vapTt)<Jt) t) p) · T) µ x = , x e o , 1t . cruv x 4 i) Na IJpt:Od tt f' . ii) Na ppt:Od p(a napayovaa Tt)i;

g( )

(

r( )

g ( X) =

cruv

)

[ ] [

GTO o ,

y) A(vt:-rm 11 <n>vapTt)<Jt)

(

)

g ( x)

1 x 20 7

AU<1ll a) i) fta Ka0E

•

]•

f ( x ) = x 201 8 In x ,

x e O , + oo .

i) Na IJpt:Od 11 f' . ii) Na ppt:Od

pia napayovaa In x a-ro 0 , + oo .

x f' (x) = cruv( 3 x + 1) - 3 x ri µ (3 x + 1) . ii) 'ExouµE f'(x ) = cruv( 3 x + 1) - 3 g( x ) . e

lR :

Apa, g ( x ) = .!. cruv ( 3x 3 Mia napfryoucm TT)�

G (x) =

+ 1 ) .!. f ' ( x ) . -

<HO

i- 11 µ ( 3x + 1 ) - � x cmv ( 3x fta Ka0E x [ 0 , :]

fta

Eivm 1) . Eivm

•

TT)�

Ka0E

( 0 , + oo )

TT) �

f : lR ...+ lR

'Ea-rro

pt: rino

f ( x ) = ( 3 x 2 + 2 x + 1 ) ( x - 1 ) 2° 1 •

Na ppt:Od pia napayovaa Tt)i;. Al><1l)

x-1 =y. TO'tE x=y+ 1 Km f( x) = [3(y+ 1)2 +2(y + 1) + 1} 2011 = (3y2 +8y +6)?11 = = 3Y°19 +8y2018 + 6y2011 = 3(x-1}2°'9 +8( x-1}2°'8 + 6(x-1t11 Mia napayoucra f crw A.om6v Eivm cruvapnicrri : 20 018 2 ( x ( x -1 ) ( x -1 ) 1 9 -1) F( x) = 3 · 2020 + 8 · 2019 + 6 · ---'--20-18� lR

f ( x ) = x2 � ,

Aivt:T«l t) <n>VUpTt)<Jt)

f'( x) = 2018 x 2017 ln x + x 2018 -x1 = 2018x 2017 ln x + x 2017 • i)

on6'tE

2020

g ( x ) cr'to [ 0 , : ] Eivm G ( x ) = .!.3 f ( x) + 3.3 E<pX =.!.3 cruvri µ�x + 3.3 E<pX .

Mia 1tapayoucra

Apa

ni�

µ 2 x · 3 cruv 2 x 3 cruv 2 x -2cruv 4 x = f, ( x) O'l.N 4x + ricruv6x = cruv6x = cruv3 4 x - cruv2 2 x = 3 g( x) - cruv2 2 x ·

2 1 i i) Apa, g ( x ) = .!. f ' ( x ) + 3 3 cmv 2 x

1 f ' ( x ) = 20 1 8 g ( x ) + x 20 7 ,

1 f' ( x ) - -1 x 20 1 1 . g( x ) = -2018 2018 Mia napayoucra g cr'to ( 0 , + oo ) Eivm: G ( x) = 20181 f ( x) - (2x018201 8)2 = 1 x 20 1 8 lnx - --x 20 1 8 =-2018 ( 2018 ) 2 ii)

'Ecr'tro Tt)i;

)

(

Iroavvqi; T aapnaA.l]i;

Eivm

x � 1.

pt:

Na anood�t:Tt: o-ri tt f txti pia napayovaa F Tt)i; onov P( x popq>l]i; F ( x = P( x 1tOA\>(l)V\)ptK1] <n>VUpTt)Gt) TpiTOt> paOpou, Tt)V onoia Km va ppd-rt:. E t<1UyroytKO l:xoA.to :

)

) F"=1 ,

)

A<pov rt f(x) eivaz (JVVeXff<; OTO

� = [ 1, +oo),

Bo. txez

G '(x) = f (x) KO.Be x � '!t:.X. G 0 ( x) = J t 2 .Jt=ldt, x ;::: l . E�illov e'!t:e1Jff F ( x ) = P ( x ) � eivaz (JVVeXff<; OTO L1, yzo. vo. eivaz o.vrff '!t:o.payovao. VJ<;f(x) OTO L1, Jyt/..o.Jff yzo. vo. urxvez F ( x ) = G ( x )+c • yzo. 1eO.Be x � , o.pKei vo. txovµe: F'( x ) = G' ( x ) , Jyt/..o.Jff F'( x ) = f ( x ) , OTO eawreplKO wv L1, Jyt/..o.Jff OTO ( 1, +oo) ( 1) . '!t:o.payovao. G(x) OTO L1, J11/..o.Jff (J1)vapVJ<J11 G(x) rtrozo. WOTe E

Bo. v'!t:apxez yzo. ·

•

F (I) G 0 (I) = 0, 0a tivm F(x) = G0 (x) yta Ka0t x e [l, +oo) .

Entt8ft µaA.to"ta

811A.a8ft EYKAEIAH:E

•

B ' 104 T.4/54

c= O,

Ma9qµaT1KQ y1a 'T'IV r AllKtto\l AuaTI

�

napayroyiaip11 pt

(0 , +

fta va tcrxl}tt TI ( 1) apKEi

= x2 � , 2 x-1 ti � P'( x ) + P (x) = x 2 .Jx - l, 2( x - 1 ) (X fi P' ( X ) + ) = X 2 , yta Kaee X E ( 1, +oo) . 2 x - 1) fta va tcrxl}Et TI 1tapa1t6.vro tO'OTil'ta apKti 'tO P ( x ) va ElVCll TI'l� µop<pfi�: P( x ) =( x-1 ) ( ax2 + Px +y ) KCll ( ax2 + Px+ y ) +( x - 1 ) (2ax+ P) + ax2 + Px +y =x2 , TI, 2 3y 7a 5P , 2 x 2 + 2 - 2a x + 2 - p = x 2 , y1a Kaet x>1. 3 5 ApKEi A011tOV 7 a = 1 ' P - 2a = 0 ' r - p = 0 ' 2 2 2 8 16 2 n, a -- A -- ' y 7 , p 35 105 P'( x ) · � +

(

)

(

)

KUl

1tClf)QO'TCl<rq.

f' ( x ) * o

Til e; v 'I £<pCl1tTOJ1tV'I O'f: 01t0l0

yia Ka9t <JTo "f f)Cl<j)lKll

( )

{

M x0 , f x0)

ol]noTE aqpdo

Ttpvti TOV Ox at aqpdo

Til e; C ,

Cl1t00£\;tTt T11V lO'OO'VV«pia. «To

OA f ( x ) = a , onov a * 0 x

dvm ptao TO'V

TOTt va

( X o , O)

av Km povo av

».

Auari

)

'Ecr'tro ( x 0 , 0 'to µfoo 'tou weuypaµµou 'tµfiµaw� OA ea a1tOOEi�ouµE O'tl f ( x) = a yta Kaee

x e ( 0, +oo) µt a # 0 . H

(

E<pa1twµEvri cr'to M x 0 , f ( x 0 ) ) Eivm

= f' ( x o ) ( x - X o ) · - f ( X 0 ) = f' ( X o ) ( X

(E): y - f ( x o ) A E ( E ) => y

- Xo) �

O- f { x 0 ) = f' { x 0 ) · { x A - x 0 ) => => X 0 · f' ( X 0 ) - f ( X 0 ) = f' ( X o ) · X A => f{ X o ) => XA = X o B 'Tp01tOc; f' ( x 0 ) 'Ecr'tro .Jx - 1 = y � 0 (1) T6'tE A<pou 'tO ( x 0 ' 0 ) ElVCll 'tO µfoo 'tOU OA, ea ( 1) => x - 1 = y2 => x = y2 + 1 => f(x 0 ) . 2 x 0 = XA => 2x 0 = X 0 f ( x) = ( y2 + 1 r y = ( y4 + 2 y2 + 1) y = y5 + 2y3 + y = EXOUµt: => ( ) x f' 0 I 3 = (x - 1)25 + 2(x - 1)2 + (x - 1)2 Mia 1tap6.youcra => X · f' ( X ) + f ( X 0 ) = 0 , Kaec yta o o TI'l� f A.ot1t6v tivm TI cruv6.pTI'lO'fl: x 0 e (O , + oo) . 3 7 5 ) (x - 1)2 (x - 1)2 (x - 1)2 Apa, x · f' ( x) + f ( x ) = 0 , ym F(x 7 +2 5 + 3 2 2 2 x e ( 0 , + oo) =>( xf ( x ) ) ' = O => xf ( x ) = a => f ( x ) = a x = 3.J(x - 1 } 7 + �J(x - 1) 5 + 3.J(x - 1 } 3 = yta Knee X E ( 0 , + ) . 7 7 3 Ilpo<pavro� a * 0 OlO'tl Ota<poptnK6. ea EixaµE = 3.(x - 1) 3 + �(x - 1) 2 + 3.(x - l) rx=J. = f ( X ) = 0 yta Kaec X E ( 0 , + oo) 01tO'tE f' ( X ) = 0 7 5 3 ym Kaee x E ( 0 ' + ) 1tp6.yµa 6.'t01tO. = (x - 1) 3.(x - 1) 2 + �(x - 1) + 3. rx=t = AVTl<JTpO<pffic;: 7 3 5 a = (x - 1) 3.x 2 + !_x + � rx=J. . Apa Av f ( x) = x yia Kaet x e ( 0 , + oo) , µE a # 0 , 7 105 35 'tO'tE TI E<pa1t'tOµEvrJ TI'l� c O''tO M ( x 0 ' f ( x 0 ) ) p ( x) = ( x - 1) 3. x 2 + !_ x + 7 35 105 Eivm TI ( E ) : y A - � = - --;.. ( x A - x 0 ) , 01tO'tE Emf3ef3auhure OT'f/ avvexeza µe /30.<Jr/ WV opuJµo Tf/<; 'lT:apaychyov 6rz 17 F(x) 'lT:apaywyi(eraz KW uro x0= 1, J17A.aJ� 6rz F ' ( 1) = 0 = f ( 1) yza tvav a.Mo rp6'1T:o A.ixl17<;.

•

[

[ (

3. 'E<JTro

(

pia

)

]

�)

cruvapT11 <rf1

f: ( 0 , + oo ) --+ 1R

EYKAEIAID: B ' 1 04 T.4/55

= XA - X o XA = 2 xo . Apa, A ( 2x 0 , 0) , 011/...aoft 'tO (x 0 ,0) eivm t0

�

µfoo 'tou OA. 4.

'Eo·tco F 11 napayo\loa TI}� f( x) =

Na anootix8d OTl F ( O) = O .

�)

� OTO l+x

1R .

F dvm ntpiTn\ Km

'EOTro h (x) = F(x) + F

( �}

f.1£ x > 0 . Na

anood;tT£ OTl 'I h dvm <JTa8tpl] (J1.)VclPTIJ<J'I Km va ppt8d TO lim F ( x) x -+ + oo

'EOTro

•

( ; ;) ( ; ;)

ppt8ti F(t) , F

1t .

( }J)

q> ' ,

( ../3) , F

b( x ) = 2

g (x) = eq>x , x e - ,

q>( x) = (Fo g )(x) - x , µt x e - , va

h ( x) = c , 6nou c ma0epa, 811/...a8ft

Apa,

A e (e) � 0 - � = -� (x A - x 0 ) �

1.l1toA.oyiotT£

Km va anood;tTt OTl

F (x ) + F

( � ) =c

Ene18ft 11 F eivm napaycoyi01µ11 cr'to 1R 0a eivm O'UVf:Xlt� O''tO JR , apa Kat O''tO X 0 = 0 , 811/...a8ft

lim F (x) = F(O) = O . x --+0

y) � X--++<xl lim F (.!.) = 0,

lim _!_ = 0 Kat limF( AUa X--++<xl =0 y...+0 X onb'te

()

F( x ) = c - F _!_ X

:::::>

X lim --+ + oo F( x) = c .

y ) 'Exouµe cp'(x) = F' ( g(x) ) · g'(x) - 1 =

1 -1= 1 . 1 -1 = cruv 2 x 1 + ecp 2 x cruv 2 x = cruv 2 x · 1 2 - 1 = 1 - 1 = 0 . cruv x =f ( g( x ) ) .

Apa, cp( x) = c , 6nou c crm0epa. Dµco� cp(O) = 0 �c=O .

Na anooti;tTt OTl 'I £1.l8da y = 21t dvm Enoµtvco�, (Fog)( x) = x , 811/...a8ft F( ecpx) = x, op1�0VT1a ariµ1t'T©TIJ Tll� C F oTav x � + oo yta Ka0e x e Apa: F(l) = F ecp = , Km va ppt8d 'I t;ioro<Jll Tll� t<panToµtvq� Tll� C F OTO x = O . = F ecp = F ( = F ecp = F

( � �J ( :) : ( �) � , (�) ( �) � . ·

J3)

Auari

a:) 'Ecr'tco P( x) = F( x) + F(- x) , µe x e 1R 'tO'te

P ( x) = F'( x) + ( F(- x ) )' =f(x) + F'(-x)(-x) ' = fta x = 1 11 crxeO'll (1) yive'tm: h (l ) .= F(l) + F(l) = 2 41t = l1t . 1 1 = f(x) - f(-x) = =0 l + x2 l + x2 n Apa, P ( x) = c , 6nou c, crm0epa. Apa, c = � , ono'te h ( x) = yta Ka0e x>O. 2 2 Dµco� P(O) =F(O) + F(-0) = 0 . Apa, yta Ka0e x e JR , icrx0e1 P ( x) = O , ono'te n o) Ene18ft lim F( x) = c = 11 eu0eia = n . F(- x) = - F( x) , 811/...aoft 11 F eivm nept't'tft. X --+ + oo 2 2 Eivm op1�6vna acrl>µn'tCO'tll 'tll � C F crt0 + oo . = H e�icrCOO'll 'tll � ecpaITTo �� crt0 x 0 = 0 eivm: x) =f( x) + P l Cxouµ< : Ii( x} - F ( 0) = F' ( 0 ) { x - 0) , 011A.aoft - 0 = l · x Kat 1 f 1 = 1 - 1 · 1 f(x) - � � l + x2 � 'tf:AtKa x. 1 1+ x2 1 =0 1 - -= -l + x2 l + x2

= F( {��)) F(�H) ()

EYKAEIAHI: B ' 104 T.4/56

Ma811pa1't.KO. yia 1'1}V r ' A1>Kdo'U

Tei�" :

Auo EiravaArtirTI Ka et�aTa

0tµ a l

'Ecnro <n>VUPTll aTI f : R � R 1ia T11V onoia l<JXVE\ f 5(x) + f3(x) + f(x) = x ( 1 ) "(Ul Ka9 t:

xeR Na od;t:TE OT\ 11 f dvm "(V11 Giro� av;ovaa GTO R

p) Na ppdTt: T11V avTiaTpo<p11 Tll� f, 011>..ao'll T11V

-1

y) Na od;t:TE OT\ 11 f t:ivm <n>Vf:Xll� cno o)

Na ot:\;t:Tt: OT\ oi 1pa<piKt� napaaTaat:� Trov auvapT'll at:rov f KU\ f txovv flOVaO\KO KO\VO OTl flEio , TO 0(0,0) t) Na ot:\;t:Tt: 6Ti oi 1pa<piKt� napacnaat:� Trov f Km f £<pcl1tTOVTCll. �) Na flEAETllGETE T11V K1>pTOT11Ta Tll� f GT) Na ppdTt: TO Eflpaoov TO'l> xropiol> 1t01> nt:pimit:Tm ano T� 1pa<piKt� napacnaat:� Trov auvapT'll at:rov f Km f Km T11V t:vOda -l

-l

"' = -x

-l

+4 .

A7TUVTfl<J'I

-l

g(x) = x5 + x3 + x g' (x) = 5x4 +3x2 + 1 > 0 ri g

0ecopouµe TI'} crovapTI'}<JT) µe x Tote yia Ka0e x . :Euvenro� dvm yvricrico� Clu;oucrCl crt0 Ka.t (l)�x 1=g(f(x1)), x2=g(f(x2)), on6te x 1<x2�g(f(x1 ))<g(f(x2)) � f(x 1 )<f(x2). ApCl f eivm yvricrico� Clu;oucrCl crto Ot f, g sivm yvricrico� Clu;oucrs� crto . ApCl Eivm ClVttcrtpeq>stm. 3 lim g(x) = lim (x5 + x + x) = lim x5 = Ka.t lim g(x) = lim (x5 + x3 + x) = lim x5 = +oo Km mpou crovsxfl� co� noA.ucovuµucr1 Ka.t yvricrico� Clu;oucrCl crto Ag = R 0Cl dvm g(Ag) = ( lim g(x), lim g(x)) = R , 01t0te KCll Ag_, = g(Ag) = R . 'Etcrt yta Ka0s x icr;(l}tt g(f(x) = x � f(x) = g- 1 (x) ,0Cl sivm g-1 (Ag) =Ag =R Ka.t f(Ar) = g-1 (Ag)= R . 'Exouµe f( x) = "' � x = r-1 ("') µe x,"' e 9t onote ClPXtKiJ crxt<TI'l yivtta.t 'I' 5 +\j/3 +'I' =3 r-1 ytCl Kcl0e 9t Ol'JAC18fi r-1 (x) = x5 + x + x µe x 9t a)

E 9l

9l .

X-+ -«>

X-+ +oo

X-++oo

X-+-00

l'l

"' E

X-++oo

-oo

('1'),

ApKd VCl 8si;ouµe Ott limf(x) = f(O) . ftCl x = 0 Cl1t6 TI'} 8ocrµtvr] <JXE<JT) 1tClipvouµs 3 f5(0) + f (0) + f(O) = 0 � f(O) · (f4 (0) + f2 (0) + 1) = O � f (0) = ,mpou f4 (0) + f2 (0) + 1 0. ftCl x 0 txouµe: f4 (x) + f2 (x) + 1 � 1 � 0 < f4 (x) + 1f2 (x) + l < 1 Km

x -+0

�h�

f(x) · (f4 (x) + f2 (x) + 1) = x � f(x) = x · f4 (x) + 1f2 (x) + l , on6tt lf(x)l = l x · f4 (x) + 1f2 (x) + l I = l x l ·I f4 (x) + 1f2 (x) + l I s l x l , €tcrt 0Cl sivm - l x l s f(x) s l xl . Enei8fi lim(-l x l ) = liml x l = 0 Cl1tO Kpttfipto 1tClpsµpoA.fi� 0Cl sivm Km limf(x) = 0. :Euvsnro� f crovsxfi� crto x = 0 . o) fao Ar n Ar_ =Rn R = R, 01 s;icrrocrs� f\x)=x Ka.t f(x)1 = x eivm tcro8UvClµe� mpou I1(X)=X�X=f(x) 01t0te A.i>VOvtCl� TI'}V f"1 (x)=X, ppicrKouµe: x5 +x3 +x=x�x3(x2 + l)=O�x=O ApCl ot Cr Ka.t Cg txouv TI'}V 'I' = x evCl µovo Koiv6 <JT)µdo to (0,0). 'Ecrtco on txouv Ka.t Kotvo <JT)µtio eKto� Til� = x , tots 0Cl unapxs1 p e 9t f(p) = r-1 (p) KCll r-I (p) p . AMCl l'J f sivm yvricrico� Clu;oucrCl mo R, onote r-1 (p) <p�f((f-1 p)) < f(p)� � p < f(p) p < r-1 ( p) atono. Oµoico� fi r-1 (p)> p �f((r1p))>f(p)� p > f(p)� p > r-1 ( P ) x -+0

x -+0

X -+0

µe

aA.A.o

µe

�

atono. ClApCl 01 Cr Ka.t 8ev txouv <JTI'}KotvoV <JT)µtio eKtO� 1t0 ClUtcl 1tOU CgppicrKOVtCll = ttA.tKa txouv µovCl8tKo K01v6 <JT)µtio to (0,0). 1 8riA.Cl8fi f(0) = r- (0) = 0 'Exouµe: (r 1 )' (x) = 5 x 4 +3x 2 + 1,x e9t l/f

X ,

1 (0) l�lim r-1 (x) = r �(!1 )° (0)=1 �lim r-1 (x)x x-0 . Em<JT)� mpou f crovexfi� crt0 x = O 0Cl dvm lim f(x) = f(O) = 0 Cl1t6 TI'}V un60e<JT)

x -+0

x -+ 0

EYKAEIAllI: B ' 104 T.4/57

x -+O

AMcl

Ma9qµa'TtK6. yta TttV r · A\lKtlO\l

npoidmtet 6n

C1-•

x * 0 => f ( x )[ f4 ( x) + f 2 ( x) + 1 J * 0 => f{ x) * 0 Kovta crto x 0 = O A.01n6v, 0a exouµe: f(x) f(x) = 1 A.(x) = f(x) - f(O) = = x x r-' (f (x)) r- 1 ( f { x)) ' f { x) aUa limf(x) = O Km C 1 -( y-) = 1 => lim C 1 (f( x)) = 1 => limy-+0 y x -+0 f(x) X-+O

=! = 1 => f'{O) = 1 => limA.(x) 1 x -+o 0tµa 2 . 'Exouµe A.om6v ( (0) = (C 1 )' (0) = O Km 'Enro tt napayroyiaiptt G\JVUP'f11 att f : R. -+ R. , tt f(0) = r- 1 (0) = 0 . Apa 01 cf Kat Cg exouv Kotvft 01tO{« 1.KUV01tot.d T� «JXfot�: eq>amoµEvr) crto (0,0) T11V eu0da f(O) = l \jl - 0 = l · ( X - 0), 011 J.aoi) tTIV \jl = X f(x) > 0 yia KU8£ x E R. /;) 'Exouµe: (r1 )"(x) =20x.3 +6x= 2x(l0x2 + 3),x e91, , (l + e• )f(x) on6te (r')"(x) >O<::>x >O KUt (r1 )"(x)<O<::>x <O yia .Ka8t x e R. ( 1 ) f (x) f( ) + x 1 Apa 11 f- 1 eivat KOiA.11 <JtO ( -oo, 0] Kat KUptTJ <JtO ((. Na anoodl;nt OT1. f (x) = ex , x E 1R K«l vu (O,+oo) • • •

-00

-too

( �)

1 (/- ) " (x) 1 KoiATJ KU PTil a +32f3 < In ea +32e� 1�· Na anood!;£T£ OT1. <JT) 01 Bt>0dec; 'I' = x Kat \j/=-x+4 eivm Ka0etec;. yia KU8£ a, 13 e R pt a<fl. E�illou r- 1 ( x) � x ' yta Ka0e x E [O, +oo) 016t1 11 y. Na t>1tOMyia£T£ TO tpfl«OO TOt> xropiot> 1t0t> 1 f - eivat KUptTJ Kat 11 \jl = X eivat Eq>a1ttOµf:Vll ntpud.dnm ano ni yp mpiK'I\ napanaatt ni� Tile;. To «=» 1crx\m µ6vo y1a x= 0. G\JVUP'f11 att � f ,niv napaflo�;q y = x2 + 1 Km 0

CJ.K.

A6yro Tile; cruµµetpiac; trov cf Kat Cg roe; 1tpoc; tTIV 'I' = x to �11w6µevo eµpaoov 0a dvm icro µe to 011tAa<JtO tOU eµpaoou tOU xropiou 1tOU 1tEptKA.f:if:tat a1tO tTIV Cf- I tTIV \jl = X Kat tTIV "' = -x + 4 . fao [0,2] exouµe:

r- 1 (x) = -x + 4 <=> x 5 + x 3 + x = -x + 4 <=> <=> x 5 + x 3 + 2x - 4 = 0 <=> <=> (x - l)(x4 + x 3 + 2x 2 + 2x + l) = 0 <=> x = l Kat -X + 4 � X (ytati;) I 2 Apa E=2 J (C 1 (x) - x)dx + 2 J (-x + 4 - x)dx I 0 I 2 = 2 J ( x 5 + x 3 )dx + 2 J (4 - 2x) = I 0 I 17 x6 4 X = 2·[ 6+4 l o + 2 · [4x-x 2 Ji2 = 6

(

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X=1. AmiVTfl<JTJ a) { l) => f ' (x) + f(x) f '(x) = f (x)( l + e x ) =>

'f11V £t>8da

f'(x) + f , (x) = l + ex => (ln f(x) + f( x )), = (x + e x ), -f (x) => ln(f(x)) + f(x) = x + e x + c. (2) fta. x=O txouµe f (0) + f(O) = 1 + c => 0 = c

I:uvemoc;

( 2) => 1n f(x) + f(x) = x+ex => lnf(x)+f(x) = ln ex +ex => h { f(x)) = h(e x ) (3), 6nou h(x) = ln x + x

AXAa

h' (x) = _!_x + 1 >

yia Ka0e x>O, on6te 11 h

eivm 'YVTJcriroc; a.u�oucra., cruvemoc; Km « 1-1 ». Apa. (3) => f(x) = ex yta. Ka0e x e lR . •

KaT«Kopt>q>t� aa6p1tTroTt;

EYKAEIAHI: B ' 104 T.4/58

Ma811pa'TtK6: yta 'T'IV r ' AVKElo\l

x 1 < x 2 � (( x , ) < (( x 2 ) an6 6nou npoK61t'ttt on f( a ;2p) -f(a) 3 f(p) - 3 f( a ; 2p ) p-a>o 3 --=---� < 2 -a -a P P x --.o- x a + 2 PA ) < f(a) + 2f(P) � 3 e a+32P < ea + 2e13 1 3f( . lim =0. 3 x1--+im0+ -X = +oo Km x --.o- g(x) a+2 p a+2P e a + 2eP lim eY = l i m g ( x) = +oo . Apa TI tuetia +oo on6tt e 3 < lne 3 < � � y--+-+«> x --.o• 3 x=O tivm KataKopU<pT) acri>µ1ttCO'TT) 'TT) i; Cf npoi; ta a a ln( e + 2e13 ) � a + 2 P < ln( e + 2e13 ) . mivro Km an6 aptcrttpa 'TT) i; . 3 3 3 Acr6p.mroTE� Tll� p.opqn\ � y=>..x+p E>tropouµt O"UvaptT) O'TJ 'tTJV y) (x) g � x 'Exouµt: = e . Alla X--++<X> lim .!... = O Km Eivm h(x) = f(x) - x 2 - l = e - x 2 - l, x e !R. X X x x h(O) = 0 Km h(x)=e -2x,h' (x)=e -2, h<3> (x)=ex > 0 g(x) = l � A. = 1 � limeY = lim l � yia Knee x E :JR . Apa T) tivm yvricriroi; au�oucra y--.o x x Icrto :JR . EniO'T)i; h" (ln 2) = 0 Km to np60'T)µo 'TT) i; -I e x --1 = (Mop<pfi h ( x) Kaeroi; Km TI µovotovia TTJi; h' = A.x (x) (xe (x) g x - x) = <p 1 a1tElKOVitOVtal O'tOV napaKatro nivaKa x x +oo ln 2 O , ,r + 0 _ opiou ). E<papµo�ttm 1Wt1tov o Kavovai; e h " (x) 0 '\i ?' e ; - 1 e; l x ' e; . H h napoucr1ai;e1 0A.1K6 eA.axicrto mo ln 2 to Ho sp ital.'E<nro <p, ( x) = = 1 1 ' = 2(1 1n 2) > 0,0"Uvenroi; h (x) > O yia Kaet h'(1n2) 2 x x X E :JR ,01tOtE T) h yvricriroi; au�oucra O'to :JR . 0 Tott X-++«l lim <p1 ( x) = e = 1 � X-++«> lim <J>( x) 1 � p 1 Enoµ£vroi;: Apa TI wetia y=x+ 1 tivm nA.fryia acrl'.>µ1ttCO'tTJ crto x > 0 � h(x) > h(O) � h(x) > 0 � f(x) > x 2 + 1 (+ oo ),oµoiroi; ppicrKouµe 6tt tivm Km crto ( oo ). Km x < O � h(x) < h(O) � h(x) < O � f(x) < x 2 + 1 . x p) 'Exouµt: f'(x) ( (x) = e > O;yia Knee MovaOtKO Aol1tOV KOlVO O'T)µtio t(l)V cf Km 'tT)i; x e :JR ,O"Uvemoi; TI f tivm KUptfi Km TI f yvricriroi; napapoA.fii; eivm to (0,1). To /;T)touµtvo tµpaoov 'Exouµt A = IR * Km g ( x) = xe� , on6tt µovaoucft nieavft KataK6pu<pT) acri>µntro'TT) tivm TI x=O. 'Exouµt lim .!... = -oo Km lim eY = 0 . Apa g

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au�oucra O'to :JR . e.M. T. yia 'tT)V f O'ta oiacrtfiµata [a, a + 2 P ] Km

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3 a + 2 a + 2 , P Km [ --P p ] ea U1tapxouv X 1 E a, 3 3 a + 2 P pA tEtota , COO'tE , X 2 E --, 3 f( a + 2P ) - f(a) 3 f( a + 2P ) - f(a) 3 = f '(x , ) = a 3+ 2 P -a 1 · p-a 3 f(P) - f( a + 2p ) 3f(P) -3f( a+ 2P ) 3 Km f ' ( x2 ) = � 2 a P � P- 3

(

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ea tcroutm µt to

E = J� (f(x) - x 2 - l)dx = J� (e x - x 2 - l)dx = - x I = 3e; 7 < +

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B' 104 -r.4/59

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Ma&r)µanKa '}'\O ntV r· At>Ktiot>

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f' KroCY'Ta BaKaMno1>1.ov

(

1 (A + B = 0 Km A - B = 1) , it ( 2A = 1 Km B = -A) , L\(vnat 'I m>VUpTl)aq f( x ) = -2-1 x 07t6ts: it A = Km B = A) Na ppt:Od 'TO nt:ofo opiapo-6 TI)� Ar . B) Na ppt:Oo'6v ap10poi A Kat B roan, 1 = -A + -B = + f(x) = ,1 = -l x l x l x -1 x2 - 1 x - 1 x + l 1 1 r) Av A = - Kat B = -- va p pt:('Tt: pia apx1KT1' r) Mta apx1Kft tTI<; f crto ouicrtTlµa ( -1, 1) Eivm 11 2 2 cruv<iptT1cr.., F µs t'67to: TI)� f CY'TO 01.UCJ'T'lpa ( -1, 1 ) . F(x) = ! [1n (l - x) - ln (x + 1)] + 2017 011A.aoit 11 : A) Na ppd'Tt: 'TO t:ppaoov 'TO'l> xropfov n 1tO'l> 2 1tt:puddt:'Tat ano T1) ypaqmo\ napaa'Taaq F ( x) = ! In l - x + 201 7 1 2 x+l ' x = -2 TI)� f 'TOV «;ova x 'x Kat 'T� w0£1£� A) fta Ka0s xe · ( -1,1) , f(x)<O to �11touKai x = -1 . 2 E) Na ppt:l'Tt: 'Ta nt:o(a opiapo-6 'TIDV 2 I =-[F(x µsvo sµ�aoov eivm: E fl ) = -f x dt J 2 { ) = J ( ) avvapn\at:rov: F1 (x) f(t)dt , 2 I

A<JKTt<JTt I :

Z) 0)

I2+2

e'2 +I

Na l..vOd 'I av(aroaq:

x2 + x+ 2

f f(t)dt 0 <

x1 + 2 x+4

Na od;t:'Tt: O'Tl 01 1::;1aroat:1�: ax 2 - a - 1 = 0 Kat f(x) = a txovv 'TO ioio a'6vol..o l..'6at:rov, (npo<pavro� KaOt: pia t:n1>..v op£V11 CY'TO a'6vol.o opiapo-6 TI)�) Kat va ppd'Tt: 'TO nl..l} Oo� Kat 'TO npoaqpo 'TIDV l..'6at:rov TI)� 1::;iaroaq�: ax 2 - a - 1 = 0 , y1a 'T� oiu<popt:� 'Tipt� TI)� napapt'Tpov a e lR Na o>.otl11proat:'Tt: T1) pt:liTI) TI)� avvapTl)aq� f Kat va KUVt:'TE T1) ypa<p1Kl} TI)� napaa'Taaq.

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Au<J11 :

!j =� �( � � )

( ) [� �};;

f f(t)dt Kat F3 (x) = [/(t)dt . l:T) Na l..'6at:'T£ Tl)V t:;iaroaq: f f (t)dt 0 F2 (x)

-�) .

�

To 7tsoio opmµou trt <; f Eivm:

[� H �� )) + 201 { =

[ (� �) im

+ )Off

-im

(�;:} ]

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= - !�! + !� 3 = !� 3 + !� 3 = � 3 �� 2 3 2 2 2

�e K<i0e 7tepi7ttcocr.., 7tpfae1 Km apKei to x Km to «K<itco <iKpo» tou o/..otl11proµatoc; va avftKouv crto iOio 01<icrtT1µa tou 7tEOiou optcrµou tTI<; f 011/...aoit crta : ( -oo, -1) it (-1, 1) it ( 1, +oo) . 'Etm sxouµs : A F, = ( -1, 1) A F2 = ( 1, +oo) A F3 = ( -1) l:T) E7tEtOit e x 2 + 1 > 1 yta K<i0e x E IR Km x 2 + 2 > 1 y1a Ka0s x e IR to crl>voA.o optcrµou tTI<; x2+ 2 e�icrcocr.., c; aUa Km trt<; cruv<iPtrtcr.., c; J f ( t )dt 2 +I Eivm to IR . E7ticr.., c; f(t) > 0 y1a Ka0s t > 1 o7t6ts: E)

-oo ,

Ar = IR - {-1,1} = (-oo,-l) U (-1,1) U (1,+oo) B) ApKsi va �ps0ouv ap10µoi A, B rocrts va 1crx;Ue1: -/- = � + � , it l = ( A+B)x +(A-B) , x -1 x-1 x +l E7tEtOit 11 cruv<iptrtcr.., g (x) = e x eivm KUptft crto IR (A B) , it it O · x + l = (A + B)x + Km 11 s�icrcocr.., tTI<; Eq>Cl7tt6µf:V11 c; crto cr.., µEio EYKAEIAHI: B ' 104 T.4/60

MafhtJLCX't'lKU yw 't't'IV r AllKtlO\J

A ( 0, 1) exei e�icrcoerri : y = x + 1 TJ e�iocoerri : ex = x + 1 exei µovaotid] A.uerri TllV x = 0 . Dµco�: 2 ex2 = X2 +l <:::> y=x <=> y = X2 <:::> X2 =0 <::::> X = 0 . eY = y+l y=O Apa Merri Tll � e�iocoerri � eivm 1'J nµii x = 0 . Z) E7moii x 2 + x + 2 > 1 yia Kci0e x e R Kat x 2 + 2x + 4 > I yia K<i0e x e R to O"UvoA.o opioµou Tll � aviocoerri � allci Km Tll � cruv<iptTlerri � x2+ x +2 G(x) = J f(t)dt eivm t0 R . Dµco� f(t) > 0 x2+2 x +4 yia Kci0e t > 1 . On6te µe clt01t0 anooetKVUetm on: x2+x+2 J 4 f(t)dt < O <:::> x2 + 2x + 4 > x2 + x + 2 <:::> x > -2 . x2+2x+

{

Apa Moe� Tll � avicrcoerri � eivm to O"UvoA.o: {-2,� H)

f'(x) = -2x 2 yia Kci0e x e Ar . Dµco�: ( x2 - 1 ) f'(x) > O <:::> x < -l ii - l < x < O . Apa TJ f eivm yvrioico� au�ouoa ota oiaotfiµata: A 1 = ( -oo, -1) Kat ota A 2 = ( -1, 0) f'(x) < 0 <::::> 0 < x < 1 ii x > 1 . Apa ri f eivm yvrioico� <p0ivouoa ota: A3 [ 0, 1) Kat A4 = ( 1, +oo) . 'Etcrt exouµe: ,/ f{A 1 ) = C ii� f(x), ��_ f(x) ) = (O, +oo) Otott: x llin -21- =0, xfun -1- = fun -1- ,_1_ =+oo x-+oo -+-1x -1 x2 -1 x-+-1- x-1 x + 1 ./ f{A 2 ) = c��J (x), f(O) J = (-oo, -1) ot6n: litn -1- = lim -1- · -1- = -oo , f(O) = - I X-+- 1 + x2 -1 X-+- 1 + X -1 X + 1 ./ f(A3 ) = ( X--+I lill! f(x), f(O) ] = (-oo, -1) ot6tt 1 - = lim -1- · -1- = -oo , f(O) = - I --+1- 2 xlim x - 1 x --+1- x + 1 x - I ./ f{A4 ) = ( xlim -++oo f(x), xlim -+l+ f(x) ) = (O,+oo) oion: fun-21- = Xfun1 -1- . _1_ = +oo llin-1- =0, X--+I+ X-H<O x2 -1 x -1 -+- - X + 1 X -1 Apa: to O"UvoA.o nµrov Tll � cruv<ipT11 erri � f eivm: f (A r ) = f (A 1 ) U f{ A 2 ) U f ( A3 ) U f (A4 ) = (-co,-l] U (O,+oo) . 0) fao R - {1,-1} exouµe: •

•

ax 2 - a - 1 = 0 <::::> a ( x 2 - 1) = 1 <::::> -/- = a . x -1 H teA.eutaia npo<pavro� yia x E { 1, -1} oev exei voriµa. H 1tPc0tTI yia x E { 1, -1} oev enaA.ri0eUetm, oion a · 1 2 - a - h � 0 Kat a ( -1 ) 2 - a - 1 :;e 0 . Apa exouv to ioio crUVOAO Moecov' Otci<popo ii µri ·

tOU KeYOU cruvoA.ou. fta T11V ei)peerri tou nA.ii0ou� Kat wu npocriJµou tCOV Moecov Tll � e�iocoerri � : --1- a oriA.aoii Tll � :

x2 - 1

f(x) = a ea XPTJO"tµonotiioouµe ta cruµnepcioµata

wu epcotfiµato� (H). 'Etm: ,/ Av a < -1 exei Mo aKpt�cO� etepoerri µe� Moe�, µia ow O"UvoA.o ( -1, 0) Km µia oto crUVOAO ( 0, 1) . Av a = -1 exei µia Merri tTIV x = 0 ,/ ./ Av -1 < a ::;; 0 eivm aMvatTI ./ Av a > 0 exei Mo etepoerri µe� A.uoet� µia ot0 crUVOAo ( -oo, -1) Kat µia OtO crUVOAO ( 1, +oo) . I)

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)

I:uvexisouµe Tl1 µeA.ETll Tll � cruv<iptTlerri � µe ta aKpotata, T11V KUptOTllta, ta erri µeia Kaµnfi� Km n� aO"Uµntcote� (optaKe� nµe�): H cruv<iptTlerri f napoum<isei tomKo µEyioto OtO 0 tO f (0) = -1 .

2( 3x 2 + 1) , yia Kci0e x e Ar . 'Exouµe: ( x 2 - 1)3 f" ( x) > 0 <::::> x < -1 ii x < 1 . Apa ri f eivm KUptTJ ota oiaotfiµata: ( -oo, -1) Km ( 1, +oo) f" (x) < 0 <::::> -1 < x < 1 . Apa ri f eivm KoiA.ri ow ( -1, 1) . f"(x) =

•

H O'UVclPTll erri

f oev napoum<iset erri µeia Kaµnfi�.

Ano ta 6pia wu epcotfiµato� (H) nporintet on oi eu0eie� x = -1 Kat x = 1 eivm KataKopu<pe� aO"Uµntcote� Kat ri eu0eia y = 0 eivm opisovna aO"Uµ1ttCOT11 Kat Oto Kat Oto +oo . -00

f'(x) f"(x) f

EYKAEIAHI: B ' 104 't'.4/61

-00

+ +

-1

au�oucra

Kut KU Pnl

+ -

au�oucra

<p0ivoucra

Kut KOlATJ

T.M. :

f(O)= - l

+oo + <p0ivoucra

Kat KU ptfi

- - -

�

' _}

MaOtipa-ruca yia TflV r At1Ktiot1

i - -- -i 1

--� I

' - - -2 - - - - - It - I - - j

,-•

- -

, .3

, - :-

:-2

- - - -

�

- -

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-i I I

I - - - ·

- - ii - - - 2' - - - - 1I I I : ;, - - - - -�: - - -- �: - - - · Ir - --3 - - - - Ir - -- '� ·

- - -

,-1

I -

- - - - - - - ·

;

: ., j '

--- ------

. 5-

·----

(ft)V£X1l<; Km 1ttpl'TTI} 01'0 [

- -

, : : -: - - - - - � -

- - - ·:- '

- _, - -

- - - -·- -

f' -

I - i

Li)

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;

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Evowq>tpov azoA.w

Me a<popµ� ro spdnYfµa (8) rYf<; 'lt:p OYf"/OVµevYf<; a(fl('/aYf<; ava<ptpovµe Ta 'lt:apaKa1:CJJ :

To yf:'(ov6i; 6tt Mo el;tcrrocre1i; sxouv to i.810 cri>voA.o A:Ucrerov (entA.uoµevri Ka0e µia crto cri>voA.o op1crµou 'tTIC;) oev ITT) µaive1 6tt eivm tcroMvaµei;, yev1Kroi;. fta va 1touµe 6tt ouo el;tcrrocre1i; eivm tcroMvaµei; npfae1 1tpomouµSvroi; va sxouµe Ka0opicre1 cre 1toto cri>voA.o avaq>ep6µacrte, cSrtA.aoi] 1tps1tet va ava<pep6µacrte cre eva cri>voA.o nou sxouv Km 01 ouo v6rtµa. �ta<popettKa <JUyKpivouµe Katt U1tapKt0 µe Katt aW1tapKto. Ilpoq>avroi; 1tpfae1 va avaq>ep6µacrte cr'tllv toµi] trov cruv6A.rov op1crµou trov Mo el;tcrrocrerov, Ti cre Ka1to10 u1tocri>voA.6 'tTIC; · n. X· 01 el;1crrocre1i;: -1- = 1 ' -1- = x + 1 , x2 - l x-1

1 = x - 1 , x 2 - I = I 1tpoq>avroi; sxouv to io10 -x+l cri>voA.o A:Ucrerov L = { Ji, -Ji } , e1ttA.u6µevrt Ka0e

µia crto avticrw1xo cri>voA.o op1crµou: D I = IR - {1,-1} , D 2 = IR - {l} , 0 3 = lR - {-1} , D1 = lR . fta va 1touµe 6µroi; 6t1 ouo a1t' autsi; eivm moouvaµei; 1t.x. rt 211 Kat rt 311, 1tps1tet va sxouv v6rtµa Km 01 Mo, cSrtA.aoi] va 0eroprt0ouv op1crµevei; crto 0 2 n o3 = IR - {1,-1} .., cre Ka7t010 cri>voA.o A � D2 n D3 •

ACiKl}Cil} 2 :

Aiv£Tm 'I at>VUP'f11 CJl1

f (x) =

In{ x + .Jx2 + 1 )

A) Na ppt6d TO ntoio opiaµ.oi> Tll <; f Km va fl.£U'f11 6d 'I f roe; npo<; 'f'I fl.OVOTOV{a Ta aKpOTaTa 'f11V Kl>pTO'f11Ta Km Ta CJl}Jl.da Kaµ.n'll <;· B) Na ppt6oi>v Ol p�t<; 'f11 <;, TO 1tpOCJ11 Jl.O Km TO O'i>voA.o Tlfl.IDV 'f'I<; at>VUP'f11 CJl1 <; f. f) Na anootixOd 6Ti av µ.ia at>v6:p'f11 CJ11 f dvm

, J 'TOTE i<JX(>ti:

a a

fu f(x)dx = O Na Ppt6d TO oA.otl-qproµ.a: f-2°0117 ln ( x + �x2 + 1 ) dx 2 7

J:17µdwm,.-

@a µnopovaaTe va ppeiTe TO oloK):tjpwµa avTo xwpiq va XP11mµono11jaew TO ep<hTtjµa (I');

Anood;Tt On TO tµ.paoov TO'l> xropioo n, 1t0l> ntpitltif:Tm ano T11 ypaqnK'll napa«naCJll Tll<; g(x) = Tov «;ova x'x Kai n<; w6dt<; x2 + 1 X = -1 K«l X = 1 UJOVT«l µ.£ 2 ln 1 + T.µ.. l:T) Na Ppt6d 'I t;iaroCJll T11 <; t<pa1tTOJl.f:V'I<; £ T11<; ypa<piK'll<; napa«naCJll<; Tll<; f mo 0 ( 0, O) Km 01"fl at>Vqf:la TO tµ.paoov TOl> xropiov '2 2 µ.£Ta;u Tll<; £<p«1tTOJl.f:V'I<; Tll<; ypa<piK'll<; napa«naCJll<; T11<; f Tll<; w6d«<; £ Km Tll<; w6da<; x = 1 . Z) Na xapa;tTt T11 ypa<piK'll napa01'aCJl1 Tll<; at>v6:p'f11 CJ11 f. Auari :

�,

( .Ji)

f1a Ka0e x e 1R 1crxt)e1:

)x 2 + 1 > N = l x j ;::: - x � x + � > 0 .

Apa to 1te&io opmµou TilC; cruvap'tll crTJ i; f eivm to 1R . H f cruvexili; Kat 1tapayroyimµrt crto IR µe

J x + 0t ( x + �x2 + 1 )' =

f(x)= [1n( x+ �x2 +1 ) =

(

)

1 1 . x+� l+ x x + �x2 + 1 �x 2 + 1 x + �x2 + 1 �x2 + 1 l > 0 . Apa rt cruvap'tll ' crTJ f etvm ' rvrtcrtroi; ' = � '\/X 2 + 1 aul;oucra crto IR Kat oev 1tapouma1;e1 aKp6tata crto IR . E1ticrTJi; ' - ( � )' 1 -x = 2 = f"(x) = x +l �x 2 + 1 (x 2 + l ) �x 2 + 1 f" ( x) > O <=> x < 0 . Apa rt cruvap'tll crTJ f eivm •

( )

KUp'fi] crto (-oo , O] f" ( x) < 0 <=> x > 0 . Apa rt cruvap'tll crTJ f eivm KOiA.rt crto [ 0, +oo) To ITT) µeio ( 0, f (0)) eivm ITT) µeio Kaµ1ti]i; TilC; ypaq>1tji; 1tapacrtacrT)i; TilC; f . B) Ilapa'tll pouµe 6tt f ( 0) =0. E7teilifi rt cruvaptrtcrTJ f etVat rvrtcrtroC; aui;oucra , eivat Kat « 1-1 » 01t0tf: Tl pil;a x = 0 eivm µova&ttj. E1ticrT)i; 1CJX6e1 yta •

EYKAEIAHI; B ' 104 -r.4/62

Maeq,.unuca "(\U 'TilV r A\lKtlO\l

x<O�f(x)<f(O)=O K<l1 ')'1a x>O�f(x)>f(O)=O. fta to cruvoA.o ttµrov f (A) exouµe: f (A)= ( }�� f ( x ), }�1! f ( x)) . Dµroc;: 2 +1 )] �(=MiMi) lim f{x)= lim [In( x+vx,-.,--; = U-+0+ lim In u = . (Y7t0'1'11 ott: ( ..Jx2;if u0 = hm ( x+'\/x-�+1 ) = hm X2X--"X 2 +1 = = 0 µe x + .Jx2 +1 > 0). Enicrri c; : lim X - � X2 +1 [ !?-:)] Mt lim f ( x) = lim In ( x + "x 2 + 1 (= Mt) = lim In u = (Y7tO'VT} ott: u 0 = lim (x + .Jx2 +1) = Apa: f (A)= ( ) Aq>ou 11 f eivm crovexi\c; Km neptttil tote ym Kcl0E a. > 0 Km ym Kci0E x E [-a., a.] tcrxl)et: f(-x)=-f{x) � rJ (-x) dx =-r/(x)dx ( 1) Dµroc;, J: f{-x )dx d:::J:a f(u){-du) = f/(x)dx . 'Etcrt (l) �J: f(x)dx=-J: f(x)dx�2J: f(x)dx=O � � ra f(x)dx =0. fta Ka0E XEJR exouµe: f{-x)+f(x)= 1n (-x+ �(-x)2 +1)+1n{x+.Jx2 +1)= 1n[{-x + ..Jx2;i)( x + ..Jx2;i )J = In [( .Jx2 +1 r -x 2] = lnl = 0. Apa 11 crovaptTlO'l} f 01 eivm nepttTJ1. Onote, J-2 0117 1n (x+.Jx 2 + l )dx =0 2 J-22010117 1n(x+.Jx2 + l )dx=J-22010117 f(x)dx = f 201 7 ( x )' 017 -[ 2017 f(x)dxf [xf(x)]2-0120177 - f-22017 x · f'(x)dx = x ,....,----:- dx xf(x) ] 2017 . f (2017)-(-2017) f(-2017)- [.Jx2 +1 ]_20120177 u=x+

x->-«>

u.=,

-oo

-eo+oo

•

X--+-<O

•

!?-:

X--+-<O

x --++oo

u=x+

x --++oo

u0 = lim

X -+ +«>

+oo

+oo )

x --++oo

-oo, +oo

1� )

l:lfpdwm, . - 8a µ7ropovaaµe va V1CoA.oyiaovµe

'!rapami.vOJ oA.ocl�pOJµa Kaz XOJp fr; OJ<; e��<;:

201 7 _20 1 7

20 1 7

-20 1 7

"\/ X 2 + 1

epchrrJµ a (I')

= 2017 · [f(2017)+ f(-2017)]-0 = 2017 · O = 0 To �l}touµevo eµpaoov eivm: ' I 1 f E{01 )= J J;!;i x2 + 1 dx= x2+1 dx= -1 � [1n(x+�x2+1 )l =f(l)-f(-1)=2f(1)=2ln(l+v'2) t.µ. Eneioi) f(O) = 0 Km f'(O) = 1 11 e�icrroO'T} tl}<; eq>antoµEVT}<; e eivm: -0 = 1 ( x -0) 011A.aoi) = x . EniO'T}<; 11 crovaptT)OT) f eivm KOlAl} O'tO [ 0, ) onote ym Ka0e x [ 0, 1] tcrxl)ei: f ( x) ::; x 011A.aoi) f(x)- 0. 'Etcn 'to �l}touµcvo eµpaoov eivm: E ( n )= J�l f (x)-xl dx =J�(x-f(x) ) dx = J: xrlx -J; r(x)dx = [ � 1 - ([ xf(x)J: -J: Xr'(x)dx ) = �- (f(l)- (g� dxH -m(1+J2)+ [N+11 = �-ln (1+../2)+h-1=J2-�-In (1+../2) t.µ. E)

}

-1

I:T) y

+oo

x ::;

Ano ta opm nou unoA.oyicraµe crt0 eprotT)µa (B) npol<Umet on 11 ypa.q>tKT) napacrtaO'T} tT}<; crovaptT)O'T}<; oev exei KataKopuq>ec; OUtE opt�OVtlE<; acruµntrotec;. E1ttO'T}<; oev exei OUtE 1tAaytE<; acruµntrotE<; U<J>OU:

±oo

lim f(x) � lim f '( x) = Jim 1 = 0 aUa 1 X .JX2 + 1 Jim [f(x)-0 · x] = Km Jim [f(x)-0 · x] = x--+±oo

x--+±x>

x--+±oo

+oo

X -++oo

-oo

X -+ --«>

7tpOKU1ttEl o

na aKatro nivaKa

xf' f"f

-00

+ l:.K. :

�

+oo

f (0) = 0

- - - ··· - .- •· - - - - · .. - - -: - - - - � - ·- - - - > - - - - -: · ·· - - - _- - - - - , !· - - - "- - - - - -- -

.� .

} • • .. • • L • • • • • • •. ..

. - --

;-3

-4

' /

- - _, _ /

··2

/ :

' / · - ; · "' -· ·-

.. -

EYKAEIAHI: B ' 104 T.4/63

�

;,. '

_ _ _ _

' : ' '

- -- - -- - - - � - .

·· - - .

· / ?'·· ··

-/..,. /.

·�

.. 1 .

·?

. .c�

..� . . . . . c

.. . . . .

. . . .. . .

lluo awAt� awol> c i � c 1 � y1a T I � av1ai>TflTt�

Jensen Avto:6TT)TU Jensen: Av TJ _

'·

(

Arr6oi:t�11 :

Herm ite - Hadamard .

Ano Tov Kt>ptclKo K. KaJ1no'6Ko 2° IletpaµattK6 Ai>Keto A0T)vffiv

)

01.)VclPTTJOTJ f eivm KUptft <HO foacrtT)µa �=[a,p] Km xi e [a,p], �>O yta Ka0e

A X + A 2 + . . . . . + A. v x v 1-1 ,2, . . . ,v tote.. r 1 1 2 X .

Ka i .

A 1 + A 2 + . . . . . + AV

Ei>KoA.a oianunrovouµe on:

A 1 f(x 1 ) + A.2f(x2 ) + . . . . . + A.J(xv )

-------

A. I + A 2 + . . . . . + AV

A 1 X 1 + A 2 X 2 + . . . . . + A. v x v A. I + A 2 + . . . . . + AV

y E � (ytati;).

H ecpantoµffit TT)c; Cr crto OT)µEio M(y ,f(y)) dvm T) (i:): y=f (y)(x - y) + f(y). Acpou T) f dvm KUp'tiJ 0a exouµe: f(x) �f (y)(x -y) + f(y), yta Ka0e XE � => f(xi) �f (y)(xi-y) + f(y) => �· f(xi) �f (y)( �· xi-A.rr) + �· f(y), yia Ka0e i=l ,2, . . . ,v, OTJA.aoft:

A 1 · f(x 1 ) � f ( y )( A 1 · X 1 - A 1 · y) + A. 1 · f( y ) A. 2 · f(x 2 ) � f ( y} ( A 2 · X 2 - A 2 · y) + A. 2 · f(y)

Me np6cr0eOTJ Ka-ta µeA.T) ppicrKouµe:

A v · f(x v ) � f{y} ( A v · X v - A v · y) + A v · f {y} A1 · f(x1)+ A2 · f(x2) + . . . + A-v· f(xv) � f (y)[ A1 ·X 1 + A2·X2 + . . . + Av·Xv - (A.1 + A.2+ . . . +A.v) ·y] + (A.1 + A.2+ . . ... +A.v) · f(y) => A1 · f(x1)+ A2 · f(x2) + . . . . . . + A-v · f(xv) � f (y) · O + (A.1 + A.2+ . . . .. +A.v) · f(y) => A

1 f(x 1 ) A.2 f(x 2 ) . . . . . A.J(x v ) -'------'- ------ � f(y)= A. I + A 2 + . . . . . + AV

(

A 1 X 1 + A 2 X 2 + . . . . . + A. v x v A. I + A 2 + . . . . . + AV

EtotKft nepintcoOTJ (01)vft0TJc;), yia: A 1 = A 2 = ... = A v = 1 .

•

Avu16tT)tU Hermite-Hadamard.

( P ) _p -1_ 2 ( ; P ) <p - a) 2 f

Av ri 01.)VUPTTJOTJ f eivm KUp'tft crto otacrtT)µa [a,p] µe a<p, tote: a+

Arr60tt�T) :

rt3r dx f( a } + f( (x) s

2 Ja ApKei va oi:i;ouµe on: a

fCx) dx s

[f(a)+f(P)J ( P - a)' .

= f: r (Ilp6Kettm yia Mo 01.)Vap'tftcrttc; f ( x), ( x) = f ( + P- x) 01.)µµetptKec; npoc; tT)V eu0eia x=A., 6nou A = a + p to µfoov tou [a,p], acpou f{A + x) = g( A - x ), yta Ka0e x e [a- P , p - J .

fta TT) 01.)Vexfi crto (a,p] 01.)VclptT)OT) f eUKOAa anooetKVi>etm g

crxeOTJ:

roe;

f(a + � X) dx -

2 2 fta tT)V 01.)VclptT)OTJ h(x) = f(x) + f(a+ p - X) AOtnOV, nou opi�etm <JtO (a, p] (ytati;), 0a exouµe: h(x)dx = 2 f(x) dx ' on6te apKd va oei;ouµe O'tl: 2

r( ; ) <P - a) a

s f.h(x)dx s [ f(a)+ f(P) J ( P - a), �

• Etvm 6lKaLoA.ovriµtvrptA.tov 'l ava<t>opci cnri axfori 2

On:

f: f(x)dx

�

f[ f ( x ) + f(a + 13 - x) ] dx (l

EYKAEIAHI:. B ' 1 04 T.4/64

f(X) dx

(;) ( ;P) s

r. f

Ma011µa't1Ka yia 'T1}V r At>Ktio\J

p dx $ h(x)dx $ 1: [f (a)

+ f (/J ) ) dx, � T£AtKci 6n:

h(x)s f(a)+f(f3) (1).

Ilpfryµan: 'ExouµE h'(x) = f(x) - f(a+f3 - x) Kat €1t€toi) f -!'

@a BXOUµE: h'(x) >0 <=> f(x) > f(a+f3 - x) <=> x>a+f3 - X <=> X E

(; ] 11

a f3 , f3

Kat

a+� ). 2 , . a+� a+� a+� a+� Apa: mmh(x) = h( -- ) = ( -- ) f(a+f3 - -- )= 2 ( -- ) Kat a<pou h(a) = h(f3) = (a) 2 2 2 2 ea Eivat: max h(x) = f(a)+ (f3). H oxemi ( 1 ) €tVat 1tAEOV npo<pavi)�. v TI cruvcipTIJITTJ f(x) Eivm KOtAT) oto [a,f3], t6t€ ot avto6TI)t€� aute� tcrxj>ouv µE avEotpaµtvri <popa Kat anoo€tKV'6ovtm €Vt€A©� av{J)..oya. AKoA.ou0ouv Mo E<papµoye�, µta 0€ Ka0€ avto6TI)ta h'(x) < 0 <::>x e [a,

•

1 '1

E<papµoyt} :

211 E<papµoyft :

f + f(f3)

ABf toxj>Et: T)µAT) µBT) µr s 3J32 . 2. Na od�€t€ 6n: J2 s J .J1 + 8x 3 k€ Ka0€ tpiyrovo

In( l] µAl]µB ljµr) $ In ( � J, � In ( T1µA) + In ( T1µB)+In( T1µr) s 3In � , In( T1µA) +In (�µB)+ln( T1µr) s � , ln('A)+ln( T1µ--r)....;... s ln ( T) µ-n3 ) , T), --' ---'-11µ -'- --'--11µ3--B)+ln( -- ----' ln(T) µA)+ln( 11µB)+In ( 11µr) (11µ A+B +r ) T)' f(A)+f(B)+f (r) - f ( A + B +r ) ' 3 3 3 3 6nou f( x ) In( 11 µx ) x e ( O, n ) . ft' auto apKEi f(x ) va Eivm KOtAT) ow (O,n). Ilpayµan f"( x ) --\- O mo (O,n). T) µ x EvtEAffi� avUA.oya µnopEi va anoo€tX0Ei Kat avto6TI)ta cruvAcruvBcmvB s .!. , 0€ o�uyrovto 8 Aucn:t� :

I ' E fj)a pµ O"fl] :

ApKBi

�

µE

•

f ( x ) .Jt + 8x3 , a=O Kat f3=1 , 24x ( 1 + 2 x 3 ) acpou f(x) cruvExi)� ow [0, 1 ] Kat, f "( x ) 0, yta Ka0€ x e ( 0, 1 ) , OT)A.aoi) f(x) ( 1 + 8x 3 )vl + 8x3 f (a) ; f( � ) .[I �-19 2 . KU pnj cno [O, t ] K t f ( ; � ) � +s(H ..fi. ,

tpiyrovo (k€ µT) o�uyrovto dvm npo<pavi)�). 211 E<pap µoyft : IlpoK'61tt€t aµEoa an6 TI)V OWt€pT) avtcr6TI)ta µ€ =

�

EYKAEIAm B ' 104 't.4/65

�

ettJaTa TiaAa10Ttpwv Ewoxwv EmµtM:ia:

rtropyoc; Ta.crcr61touA;oc;

E.M.Il. 193 1 EII:ArnrIKEI: EEETAI:EII: Ano to rrueayopeto 0erop11µa crw opeoyrovto IlOAITIKnN MHXANIKON tpiycovo HAf npoKl>n'tet Ott Af' 2 = Hf' 2 + HA 2 = 4 2 + 2 2 = 20 Km etcrt 3 o 0EMA TPirnNOMETPIAI: T11 Atf1axo� Mnal.:taafha� Af' = 2 J5 . Ma0T)µawc6<; ow fuµvamo Ktpaµetffiv Ktq>a/..ovta<;.

Ot nl.zupt<; unoA.oyicrtT)Kav, o unoA.oyicrµoi; tcov To etµa 'fttav to e�'fti;: ycovtrov µnopei nA.Eov va yivet µe neptcrcr6tepoui; Na enV..ueei rpiymvov ABI' 6o8d<J1'f� rr/� 1CAevpa� ano evav 'tponoui;. A<; 8ouµe evav . . . rov BI'=3m, rr/� 6zxor6µov rov ALJ = 2 J2 m Kaz Ano to opeoyrovto tpiycovo HAf �picrKet Kanoto<; yvmowv 6vro� orz 11 61xor6µ0� ALJ ux11µari(e1 µe AH 2 rrJV 7rJ..wpav BI' ol;dav ymviav 45° . euKoA.a Ott Eq>f' = = = 0, 5 . I:uvenroi; Hf' 4 H wpecrri tCOV ouo nl.zuprov ea yivet xcopi<; va 'Etcrt Aol1tOV XP11crtµonot11eei tptycovoµetpia, µovo µe yvrocret<; r = roi;o &rpO, 5 = 26°33 54 Eutlei8eta<; fecoµetpiai;. B= 90° + f' = 90° + 26°33 ' 54 " = 1 1 6°33 ' 54 " . A EuKoA.a nA.Eov µnopei va �peeei ott A = 1 80° -B - f' = 1 80° - 1 16°33 '54"-26°33 '54" = = 36°52'12" H eniA.ucrri teA.etrovet Kanou eoro. Na tovicrouµe Ott to 1 93 1 ot unO'lfftq>tot 'fttav ecpo8tacrµEvot µe nivaKe<; tptycovoµetptKrov apteµrov. Ot nivaKe<; autoi 'fttav unepnoMttµot Km t6vot µeA.aVT)<; eixav 8 I H 6 XP11crtµon0111eei yta va eKruncoeouv. Av etA.ouµe !::.ev �Mmetm 11 yevtKOtT)ta av unoefoouµe Ott Va xapaKtT)ptcrouµe tT)V npocrnaeeta tCOV B > t . Av cptpouµe tote to U'lfO<; AH tou µae11µattKrov yta tov unoA.oyicrµ6 'tCOV 6crcov tptyrovou, to B ea eivm µeta�U H,r Km to tpiycovo un'ftpxav crtoui; nivaKe<; autoui;, µnopouµe va HM ea eivm opeoyrovto Km tcrOcrKeA.Ei;. Acpou ypa'lfOUµe ott 'fttav avvntpfU.l)Ta Komam1K1] Kat <n>vaJla 11pro1K1] A!::.= 2 J2 , tcrx6et ott AH=H!::.=2. Ilapa6tTro Jlta aKOf11J A.i>Gl], npo<pavro� fao tpiycovo ABf tcrx6et ott B6,; = A + f' . <n>V6&TOT&p1) U1t0 &KelVfl TO\l <n>vaotl..<pot>. A A fao tpiycovo AB!::. tcrx6et ott B sr; = + 45° . A ucn r 'Exouµe: 45 ° = A Li B = - + f' Km 135 ° = 2 2 A A I:uvenroi; tcrx6et ott A + f' = + 45° � A � r = + B . ApKei A.ol1t6v va unoA.oyicrouµe 2 2 0-B-r 1 80 0 � 1 80 - B = + 45 o � B - f' = 90o . A 2 tT)V ycovia A ' Otott tote r = 45 ° - - Km B =135 ° 2 t::.11 /..ao'ft to tpiycovo ABf eivm 'V£Uooop6oyrov10 A tpiycovo. ea anooei�ouµe ott HB Hf' = AH 2 . (eivat Be npocpavij<; ttm Km 11 crxtcrTJ B t =90° Ta opeoyrovta 'tpiycova HAB, HrA eivm oµota. 2 H ano8et�11 aKoA.oueei aµEcrcoi;. CftTJV onoia �a�etm 11 npo11yol>µeVTJ MCTTJ) . ycovHBA=A+f. 'E'tcrt mo opeoyrovto tpiycovo LtT) cruvtxeta an6 to voµo 11µtt6vcov unoA.oyil.;ouµe HAB tcrx6et ott ycovHAB=90° - ycovHBA= tt<; �. y. AA.Ml a=3m, on6te apKei va unoA.oyicrouµe =90°-(A+f)= 90°-( 1 80°-B)=B-90°=r. __ , 811/..ao'ft to 2R (cre m). I:uvem:Oi; HB Hf' = AH 2 = 2 2 = 4 Km enicrri i; to A.Oyo _!!_ µA 17 Hf' - HB = Bf = 3 . H OtXOtoµo<; A!::. 8..tpxetm ano to µEcro M tou Ano eoro npoKl>ntet Ott Hf' = 4 Km HB = 1 . Ano w .nueayopeto 0erop11µa crto opeoyrovto to�ou Bf' . Av MN =2R, t6te MNJ.Bf crto µEcro tpiycovo HAB npoKl>ntet ott tT)<; K Km M A N = 90° = !::. K N = M t N. Apa AB2 = HB2 + Hf'2 � AB2 = I 2 + 22 � AB = J5 . All.KN M!::. ·MA= onote: eyyp<l'lftµo, I

•

• • •

("'\

EYKAEIAm B ' 104 T.4/66

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@tµa'Ta Ilal..<uo'Ttprov Enoxrov

MK- MN=Mf2 . Av A.01nov MK=x, ton: �=x Km M/!i=x fi. , onote: N

-------

svBda�, KW P) Na mr:oloylaBovv ra µ�Kr/ BE /\

= x KW LJZ /\

Enetofi eivm A B r = A l!i r = 90 ° , roe; eyyeypaµµevec; cre K'6tlo nou �aivouv cre 11µtK'6tlt0 µe to Ilu0ayopew E>erop11µa naipvouµe: Br = R Km rl!i = R fi. , onote A 2 = 3 0 ° Km A \J(j'l :

AK0µ11

r1 = A1 = 45 ° .

E1 = Z1 = 1 5 ° (1).

r2 = 75 ° Kat

eivm

M/!i·MA=Mf2 6 x fi. ( x fi. + 2 fi. )=x2 + Kr2 9 9 => 2x(x+2)= x2 + - => x2 +4x - - =0 => 4 4 1 1 9 1 . x= - . E�aUou MK-MN = Mf2 => - 2R = - + 2 2 4 4 3 => R= � . Apa: _!!___ =2R => - =5 => 2 17µ A 17µ A 3 11µA= - K.A.n. 'Etcrt B ' r yvrocrtec; Kat �=2R11µB, 5 y=2R11µr yvrocrtec;. Il UpU Tll P 'l (j 'l : :EtT)V iota E�lcrffiCTTJ µac; 0011yei 11 npoq>avi}c; tcrOtT)ta /!iA·�M=/!iB · /!if ( 1 ). Ilpciyµatt: (1 )=> 2 fi. · x fi. = (KB x )(Kf + x) 3 9 3 => 4x = ( - -x)( - +x)=> 4x= - -x2 => 2 2 4 -

x2 +4x - 2 =0. 0 unoA.oyicrµoc; tou x µnopei va 4 0011yi)cret Kat cre aneu0eiac; unoA.oyicrµo tTJ<; 1 A x 2 1 A aq>ou Eq> =ecpB A M=ecpB r M= Kr = 3 = . 2 3 2 A

1 956 E.M.n. noA.tnKoi M11xav1Koi (el;E'T«an\� K. Ilmrmroavvot>)

rEOMETPIA:

El� KiJKlov mcrivo� R Bswpovµsv cyyeypaµµtvov rsrpa:!!kvpo v ABI'LJ, wv 01r:oiov '1 Jzaymvw� AI' dvw JuJ.µsrpo� wv KiJclov KW dvw

Stott E B Z = E /!i Z = 90° ' apa B1 = Z1 + Z2 (2) /\

aUci Kat B1 = r1 crav eyyeypaµµevec; cre K'6tlo /\

nou �aivouv crto iow to�o, 011A.aofi B1 = 45 ° (3). Ano nc; (1), (2) Km (3) naipvouµe: 45 ° = 1 5 2 + Z2 , t1 Z2 = 3 0 ° . Ano to op0orrovt0 /\

tpiyrovo fHZ exouµe: r3 = 'Xf -Z2 , t1 r3 = 60 ° . /\

T0 a0poicrµa t(l)V yrovtc.Ov r I ' r 2 ' r 3 eivm /\

r1 + r2 + r3 = 45 ° + 75 ° + 60 ° = 1 80 ° , apa ta CfTJµEia A, r, H avi}KOUV crtr)V iSta eu0eia. p) Ano ta 0µ01a op0oyrovia tpiyrova A/!iE Km R .fi. + y R Jii . . ABZ exouµe: = r:; � , 11 · Rv3 + x Rv 2 y.J2 = R + x (4) Km ano ta 0µ01a tpiyrova Rf2 EBf Kat f/!iZ naipvouµe: 2'. = , fJ y = xJ2 ( 5).

.J3

AB = R .J3 , Ano ttc; (4) Km (5) �picrKouµe x = R ( 2 + .J3 ) Km

M = R fi. . ea 1rapa�awµsv Jza E KW z m Uf!µda wµ�� rwv wrtvavrl 'll'Asvpmv AB, JI' KW ALJ, BI'. eswpovµsv KW m� 'll'Spupspda� Jw.µhpwv I'E KW I'Z W O'll'OfW 6Kr0� WV I' reµvovrw KW Sl� n Uf!µsiov H Z17wvvrw: u) Na JszxB� 6rl m Uf/µda A, I', H KEivrw 6'/l' '

( .J3 ) . 'H µe tptyrovoµetpia x = R · 11µ75 ° = R · 11µ ( 45 ° +30° ) = R ( 2 + .J3 ) Kat y = R .fi. · 11µ75° = R .J2 · ( 2 + .J3) . y = R .J2 2 +

EYKAEIAID; B ' 104 'T.4/67

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0tpaTa IlaMitoT£p<r>v Enoxrov

2o<; Tp07t0<; (MO 'tO otl:rio eeµciTCOV TOI) Ap. IlaM.a) a) a<pou E� j_ AZ l((ll ZB j_ AE ' TO r ea eivm TO opeoKevtpo TOU tptyrovou AEZ, onOT€ 11 At ea eivm KaeeTll crTllv EZ mo H'. AUa TOT€ T(l TetpanA.eupa BfH'E l((ll �fH'Z ea eivm eyypa'l'tµa, 011A.aoi} to H'ea eivm to oei>tepo KOtvO CJ11 µeto TCOV KUKACOV µe Otaµfapou<; fE, fZ. 1\pa TO H H 011A.aoi} T(l CJll µEia A, r ' H eivm cruveueetaKa. LTll cruvexeta ano M = R J2 = A4 ' =

l((ll AB = R /\

J3 = A3 npoKi>mouv OTt Ot f = 45 ° ' /\

B r A = 60 ° , A 2 = 30 ° , A 1 = 45 ° /\

E1 = Z1 = 90 °

-------

XP11crtµonoi11eei eivm:

0 AOyO<; Of10lOT1)Ta<; OVO opoirov Tptyrovrov dvat iao<; pt: TO AOyO OVO «VTlaTO(X(l)V ypapplKIDV aTotxeirov Tov<;. 1 '1 ecpappoyi} Tou napmravco 0ecopi}µaTO<;.

L'ta oµota tpiyrova AE>H, A�E o eyyeypaµµevo<; Ki>tlo<; tou tptyrovou ABf eivm Kotvo<; napeyyeypaµµ£vo<; Ki>tlo<; nou avncrtoixei crtt<; Kopucpe<; A, A. Li>µcprova µe o,tt ypacp11Ke , o Myo<; oµotOTllta<; TCOV tptyrovrov aUTcOV eivm icro<; µe 1 . IlpOK€tT(ll yta icra tpiyrova. LUVenro<; �E=E>H. 2'1 ecpapµoyt1 TOU napmravco 0ecopt1µaTo<;.

Lta oµota tpiyrova ABf, A�E, ZBH, IE>f o Myo<; oµotOTllta<; eivm icro<; µe to Myo trov neptyeypaµµ£vrov Ki>tlrov tou<;. LUV€1tcO<; µnopei

A = 1 5° .

p) 0 unoA.oyicrµo<; TCOV x, y yiveT(ll onro<; npomQuµ£vro<; ano n<; 0µ0t0Tllte<; trov tptyrovrov ' ' Br �E BH er va ypa<pet on - = - = -- = - = ABZ ' A�E l((ll BfE, �rz. R R, Ri � M I KPO noA YTEXNEIO

2° 9EMA EIEArnnKnN EEETAl:E!lN 1 963 l:TH rE!lMETPIA. Tql..t paxo<; Mnal.Ta«Pta<; Ma0TJµmuc6� ato fuµvaaio Kepaµmov Ke<paA.ovta�.

Zi}Tllµa 2°v :

L1i&raz rpiywvov ABI' KW w sv8dw LJE, ZH KW eJ irap<iJJ.'fA.01 avrunoixw<; irpo<; ra<; trkvpa<; BI', rA, AB KW e<pamoµsvw "f<; 81<; ro rpiywvov cyycypaµµtv'f<; irspz<pspda<;. Eav R, , R2 , R3 KW R dvw 01 aKrivs<; rwv irsprycypaµµtvwv KVKA.wv irspi ra rpiywva ALJE, BZH, ref, KW ABT, rore va &1x8d ,, U70"f<;

R, + Ri + R3 = R .

� + BH + E>r � +� +�

Ano eoro 11 �llTOUµeYll tcrOTllta eivm npocpavi}<;. To KaM µe auTi}v TllV anooei;11 eivm on anooioet l((ll OTUV exouµe aUa avtimoixa ypaµµtKU crtoixeia trov tptyrovrov A�E, BZH, fE>I, ABf. A<; ypa'l'ouµe eva µovo napaoetyµa: Av 01 aKrivs<; rwv cyycypaµµtvwv KVKA.wv rwv rpzywvwv ALJE , BZH, reL ABI' dvw p" p2 , p3 , p row zax,vs1 p, + p2 + p3 = p . B 'Tpono<;

Ano TO �eA.tio E>eµatrov Aptcrteioou

naua 'Ecrtro AN u , l((ll AM ua ta avticrtoixa U'l'll trov oµoirov tptyrovrov A�E, ABf. Tote E 2 ,&_ � U a - 2 p l 2 p l _l l 2E R ua ua ua 'tU a ua

Br E>H + BH + E>r = ---� +� +� � +� +�

ea yivet XPii CTll €VO<; eeropi}µmo<; nou oev ypacp€T(ll crto crxoA.tKO �t�A.io. H an60et;i} tou rocrtocro eivm noA.i> anl..i} Km µnopei va yivet ano eva µae11Ti} µe evotacpepov yta T1l feroµetpia. To eerop11µa eivm to mxpaKatro: Av ot nM:l>pt<; oi>o Tptycl>vrov dvm napallriu<; , TOT£ T« Tpiyrova «l>Ta dvm opota.

'Etcrt ta tpiyrova ABf, A�E, ZBH, IE>f, AE>H eivm oµota. To A eivm TO CJ11 µeio TOµi}<; trov ZH, 10. 'Eva aMo eerop11µa, µaUov npocpave<; , nou ea EYKAEIAHI: B ' 104 T.4/68

Xpt\altJCC e A.c;Kllo'I 2. AivE'fm tvac; aptOµoc; a

Ka9& v 0 e

N·

oru.1avac1

Na anoo&\;&T& 0Tt unapxt1 aptOµoc; t>O, TiToto<; ro(J't't, yia ' , ux0. pX£l v e ff µ£ v ;;:: v0 Kat ( -t v 2 + v - a ;;:: £ .

l )

Ai>o'I. ea epyacnouµe µe t11 µ€eooo Tll <; ei<; frto1to a1taycoyft<;. �11A.aoi}, ea a1toOe�ouµe on 11 UPVTI CITJ Tll <; 1tpOtacni<; auti}<; eivm 'lf€U8i}<;. AJJ.,a. yta va crmµaticrouµe TllV UPVTICITJ Tll <; 1tapamivco 1tpotacni<;, ea 1tp€1tet 1tpona va T11V ypa'lfouµe <J'tl'JV au<JTll PTt µae11µantj µopcpi} T11 <;, 11 01toia eivm: e. 3e > 0, V'v E r f ' 3v E N* , v t\ Tropa, yvcop�ovta<; toU<; voµou<; Kat toU<; Kav6ve<; t1l<; Ma011µantj<; AoytKiJ<;, ppie>Kouµe eUKOAa on 11 apVllCITJ Tll <; eivm: V'e > 0, 3v0 e f f , V'v e N. , (2) ::::> 'Ecrtco A.omov on 11 1tpOtaCITJ (2) eivm aA.11ei}<;. eecopouµe €vav apt0µo e>O. Tote ea U1tapxei apieµo<; v e N. t€t0to<; rocrte va tcrx\>et: I 1 v I e yia Kaee v e N• µe v ;;:: v 'Etm, a1t6 t1l crx€cni auti}, µe v v 0 Kat v v 0 1 , ppi<JKoµe: Km Ilpocrefaovta<; Kata µ€A.11 ppicrKoµe: 2e > (V o �

;;:: Vo l < -1 r v2 + v -ex.I ;;::

( 1) v ;;:: v0 l <-1rv 2 + v-al < e. (- r v 2 + -ex. < 0 = = + l <-1r0v� +v0 -cx.l < e l <-1r0+1 (v0 + 1)2 + (v 0 + 1)-cx.l < e . l <-1r0 v� +Vo -ex.I+ l <-1ro+I (Vo + 1)2 + + 1) -ex.I ;;:: l ( -lf°+1 (v0 +1)2 + (v0 +1)-cx.-(-lf°v� - v0 + ex.I =1 (-lf°+t (2v� + 2v0 ) 1 ;;::: �.1 (-lf°+1 ( 2v� 2v0 )l - 1 = 2v� 2v0 ;;:: 4 => 2s > 4 => e > 2 . Etlfyovta<;, yta 1tapa<ietyµa, s=l , ea €xouµe to ato1t0 1>2. Apa 11 1tpotacni (2) eivm \jf€U8i}<; Kat E1toµ€vco<; 11 apVll CITJ T11 <;, 811A.aoi} 11 1tpotacni ( 1) sivm aA.110i}<;. o

0 .

cx.v = (-tr v2 + v,v =1,2, 3

I:Tllv AvaA.ucni, 11 1tapa1tavco Merri , eivm 11 a1to8ei�11, µe tov optcrµo tou opiou aKoA.ou0ia<;, on 11 aKoA.oueia: ... 8ev mryKA.ivet ae 1tpayµanKo apieµo.

:r.xoA.to .

AOKTl 011 3. Na «1tOO&\;&'t'£ O't'l 'YUl Kcl9& apt9µo £>0, 'U1tclPX£l aptOµoi><; o>o, 't'E't'OlO<; WO't'& vu l<JXVEl:

l x2 - x - 21 < & ' 'Yl« Kcl9& x

lR µt 0 < Ix -

21 < a

•

tva<; apieµo<; e>O. ea a1to8si�ouµe on U1tapxei 8>0, t€tot0 cO<JtE, yta Kaes x E R ' va tcrx(>et 11 cruve1taycoyit : O I x - x - 2 1< e M1topouµe va aval;11ti}crouµe €vav t€toto apieµo 8 crs €va 8tacrt1lµa (0, a), 01tOU a>O, yia 1tapaoeiyµa crto 8tacrt1lµa (0, 1), o1tote 0<8<1 :Ecrtco tropa o'tt yia €vav apieµo x e R icrx(>ei: O<lx-21 < 8 (<1). Tote, €1t€t8i}: Kat x ( x - 2) + 3 � x 2 3 1 + 3 4 ' ea €xouµe: x 2 - x 2 4 x - 2 . I:uve1tro<;, yta va tcrx(>ei: x 2 - x - 2 e , apKei va t<Jx6.et: 4 x - 2 � s , apKei Ai>o'I. 'Ecrtco

l x 2 -x-21 =l x -2ll x +11 - 1< l 1

Ix - 21

�

(1 ).

I + 11 = j

: . 'Etcrt, av 0ecopi}crouµe €vav apt0µo 8

I x2

µe:

x - 2 1< e

l 1

< < min { 1, �}, tote, yia Kaes

0 8

(1). Ilpayµatt ' fotco on yta €va apieµo x 1tapa1tavco ppi<JKouµe: I x 2 - x - 2 1< 4 1 - 2 1

<JUV€1taycoyft o1tco<;

1 l - 1+ <

tcrx(>si 11

t<Jx6et: O<lx-21<8. E1tet8i} 0 < 1 Kat O < � ' 4 Kat e1tet8�: 4 1 x - 2 1< 43 e , €1tetm ott

E lR

A1to8ei�aµe on: V'e > 0, 30 > 0, V'x E IR, O I - x - 2 1< e . Auto <JT11V AvaA.ucni eivm 11 a1to8ei�11. µe tov e---0 optcrµo tou opiou cruvapt11CITJ<; , ott: :r.xoJ.to .

EYKAEIAlll: B ' 103 -r.3/69

lim(x 2 - x) = 2. x --+ 2

------ Xpl]at.pti; Ent.a11 J1UVatt.i;

Aata] att.i;

1\A.yt�po. A ' AUKtiou

A(JKfl <Hl 1 .

-------

0tcopoi>µt oi>o ap16µoi>� a Km p Km l>1to6t1'0l>J1£ 01'l l<JXi>t1: a+x>p, yia K0:6t ap16µ0 x>O. Na anood;t1'£ 01'1 a ;;:: p •

on a < 13. Ton: fHx>O. 'Etm, ano tflV uno0eO'T) µe x=J3-a, 13picrKouµe: a+(f3-a)>l3 Kat O'UVenffii; 13 > 13, atono. A.pa: a ;;::: � .

J\U(Jfl . 'Ecrtro

x2 + x + ria ouo ap16µoi>� a, p e 1R , l<JXi>t1: -1 � 2a p � 2 , y1a Ka6t x +1 01'1: l a l � 2J2 . �\.(JKfl(j'l 2 .

Au(jri. 'Exouµe, yia Ka0e x e lR LUVe7tcOt;:

{

-x2 - l � x2 + ax + l3 :$; 2x2 + 2

a2 13 2::: -_-_8 a2 - 8(13 + 1) :$; 0 8 � a2 - 4(2 - 13) -< 0 -a2 + 8 13 :$; -4

___._ _,,.

{

x e 1R .

Na anood;nt

2x2 + ax + 13 + 1 2::: 0 x2 - ax - 13 + 2 2::: 0

a2 - 8 -a2 + 8 � -- :$; -4 8

Na ppd1'£ 1'a nol..'Ucl>Vl>µa A(x), B(x) Km r(x), t1'01 roa1'£ va l<JXi>t1: (1) A(x) + (x - 1) 2 B(x) + (x - 1) 2 x3 r(x) = x6 - x5 - x3 + 5x - 3 av dvm yvco01"0 01'l 1'0 A(x) 1\ dvm 1'0 µ11otv1Ko nol..'Ucl>Vl>µo 1\ pa6. A(x) <2 Km 1'0 B(x) 1\ dvm 1'0 J1110£VlKO 1tOA'UcOV'UJ10 1\ pao. B(x) <3 . A<iKfl<ifl 1

Afo11. 'Exouµe:

[

(1 ) � x6 - x5 - x3 + 5x - 3 = (x - 1)2 B(x) + x3r(x) J + A(x) (2). Aoyro trov uno0foerov, 11

(2) eivm 11 tautotrita tflt; OiaipeO'T)t; ( x6 - x5 - x3 + 5x - 3 ) : (x2 - 2x + 1) . EKteA.c>uµe tflV oiaipeO'T) autfi Kat 13piO'Kouµe 1tl1AiKo: x4 + x3 + x2 - 1 Kat unoA.c>tno: 3x - 2 . A.pa: A(x) 3x - 2 Kat B(x) + x 3r(x) x4 + x3 + x2 - 1 . EniO'T)t; exouµe: x4 + x3 + x2 - 1 = x3r{x) + B(x) (3). Aoyro trov uno0foerov, 11 (3) eivm 11 tautotrita trit; OtaipeO'TJt; ( x 4 + x 3 + x 2 - 1) : x 3 • EKteA.ouµe tflV otaipeO'TJ autfi Kat 13piO'KOUµe 1tl1AlKO: x + 1 Kat U7tOAOt7tO: x2 - 1 . A.pa: B(x) = x2 - 1 Kat r(x) = x + 1 . 'Onroi; 13picrKouµe eiJKoA.a, ta noA.urovuµa auta: A(x) = 3x - 2 , B(x) = x2 - 1 Km r(x) x + 1 nA.11pouv ni; Oo0'µ€vei; crxfoeti; Kat apa eivat ta µovaOtKU l;11to1'.>µeva. =

A(JK'l <i'l 2

( 9t1'lKoi> npoaava1'ol..1aµoi>). 'E01'co 11 t;iaco011 : (1 - 1..) x + 21..y - 41.. - 2 = 0 (1). Na anood;t1'£ 01'1: I ) rta Ka6£ A e lR , 11 t;iacoa11 (1) 1tap101"aV£l µia £'U6da £;,. 2) Ynapxt1 tva µovaOlKO OTIJ1£io 1'0'\) (Kap1'£0lavoi> £1tmtoo'U), 1'0'\) 01tOio'U 11 a1t001'a<J11 ano 1'� £'U6dt� £;,. dvm 01"a6tp1] (avt;ap1'1)1'1) 1'0l> /..). •

J\U(Jfl . I ) H e;icrroO'T) ( 1 ) eivm tflt; µopq>J1i;: Ax+By+f=O, onou A=l-A., B=2A. Kat r= -4A.-2. 'Ecrtro Ott yta evav apt0µo A. E IR lO'x()ouv: A=O Kat B=O. Tote ea eixaµe: A.=1 Kat A.=O Km O'UVe7tcOt;: 1 =0, Ut01t0. A.pa, yta KU0e A E IR ' exouµe: A "# 0 11 B "# 0 . LUVe7tcOt;, yta KU0e A e IR ' 11 e;icrroO'T) ( 1 ) naptcrtavet µta eu0eia e i. . 2) o.) 'Ecrtro on 11 anocrtaO'T) d evoi; O'T)µeiou M(a, 13) ano ni; eu0eiei; e i. eivm ave;aptfltfl tou A.. 'Exouµe : i. I (1 - A.2 )a + 21..13 - 41.. - 2 1 I (1 - t..2 )a + 21..13 - 41.. - 2 1 d i. = l + A.2 �(l - A.2 )2 + 4A.2 I 2A - 6 1 = l -2A + 2 1 'Etm, exouµe: d + I = d_ I � "' � ... � 13 2 . EniO'T)t; exouµe: 2 =

�

EYKAEIAHI: B' 104 T.4/70

------ Xp1lalpt:; EmcntµO:va&� - AaKftm:�

--------

d+1 = d0 => I 2 P - 6 I =I a - 2 1 =>I a - 2 1= 1 => ... (a = 3 it a=l) . 2

Apa, tote: (a = 3 Km 13=2) it (a= l Km 13=2). �) A \'Tl<npo<prn.;. i)

ave�aPtrttTI tou

'Ecmo on a=3 Km 13=2. Tote, onro� 13picrKoµm d>Kol-a: d

A. e lR .

ii) 'Ecrtro

11 fta napa8etyµa d0 = 1, d 2 = . 5

d A = 1 , ave�aPtrttTI tou A.

lR .

1 31-2 - 1 1 ' 5ev eivm l + A2

ott: a=l Km 13=2.Tote, 13picrKoµm on:

'I:uµnepaivouv on unapxet tva µova8tKo tetoto crri µeio Km eivm to M(l ,2). r · ,-\ u Kdou

(

M u.O tj µ flT t K {t 0 tT l K TJ S K m Tr,zv o /,oytKit i;

Xp1'1 mµi;i; n u. p UTfl fH1 <>ui;.

Ko:rr,uOuv<>rii;)

Av µta cruvaptrtcrTJ f eivm opicrµSvti cre tva m'.>vol-o A � lR Km t<Jx6et f3 (x) = A. , yta Ka0e x e A ( A. e lR ), tote 11 f eivm crta0epit crto A ( ano8e�11:). 2) Av µta cruvaptrtcrTJ f eivm optcrµSvti cre tva m'.>vow A � lR Km t<Jx6et f 2 ( x) = A , yta Ka0e x e A ( A. e 1R • ), tote 11 f 8ev eivm avaYKairo� crta0epit crto A (ano8et�11). 2 3 ) Av µta cruvaptrtcrTJ f eivm optcrµtv.., Km cruvexfl� cre tva 8tacrt11µa � c 1R Km tcrxl}et f (x) A. yta Ka0e x e t:i. ( A > 0) ), tote 11 f eivm crta0epit crto !:!.. ( ano8et�11;) !\ c; K ri o· 11 t . Mia m>VU f>Tfl GTI f dvm m>VEX,1\<; <r'TO oia<JTl)µa [a, PJ (a < p) µt: l)

f(a) = -a,

r( ; ) a

;

a 13

Kut f(l3)

-13 . AKoµa, 1) f dV«t 'Tpt:� q>opt<; 1tapaycoyicn µ11 <J'TO (a, P>

µt: f "' ( x) < 0 , yia Ka8t: x e (a,p) . Na aitoot:i;t:n O'Tt: I)

H Cr fxt:i aKptPcl><; tva iti8avo miµt:io Kaµ1t1]<;.

2) H Cr txu aKptPcl><; tva Gt)µt:io Kaµ1t1]<;. 3) Av -P < a < 0 , 'TO'Tt: 11 f txt:i 'TO itoA.i> 'Tpt:i<; pi�t:<; mo oui<JTl)µa (a, p) , t:K 'T<OV oitoirov 11 µia

'TO'VAUXl<r'TOV dvm JllKpO'Tt:pt) 'TO'V \ 1 1 cr q .

[ a 2+ 13 , ,.,] R

a + 13 2

•

[

Ano to 0erop11µa µecrri� nµit� yta trtV f cre Ka0tva ano ta 8tacrtitµata a ' '

enetat

Ott

unapxouv

apt0µoi

;

( a, ·a + 13 ) -

a + 13 2

] µe:

Ilpoq>avro�: a < ;, < ;2 < 13 . Ano to 0erop11µa tOU Rolle yta trtV f ' crto 8tacrtrtµa [ ; ' ; 2 ] ' enetat Ott unapxet apt0µo� ; e ( ;, , ; 2 ) µe f " ( ;) 0 . Apa, to crriµeio M ( ;, f(;)) eivm nt0avo crriµeio KaµJtit� trt� =

Km enet5it f" l (a, 13) ' acpou f"' ( x ) < 0 yta Ka0e x E (a, 13) ' enetat Ott to crriµeio auto M eivm to µova8tKO nt0avo CJT)µeio KaµJtit� (11 f " 8ev µnopei va µ118evi�etm cre aUo CJT)µeio). cf .

Enet8it f " l (a, 13) , exouµe: a < ;, < x < ; < 13 => f"(x) > f"(;) = 0 => [ f KU ptit crto (;, , ;)] Kat a < ; < x < ;2 < 13 => f"(x) < f"(;} = 0 => [ f KoiAri crto (;, ; 2 )] . EYKAEIAID; B ' 104 'T.4171

------ Xp1\atµt� Enurqµavat� - AaKl\at� ------:Euµm:pcivouµ£ ott to <rr) µ£io M ( i;, f(i;)) dvm t0 µova<itKO <rr) µ£io Kaµ1ti)c; TIJc; Cr . 3) 'Ecmo

J3 > 0

ott -13 < a < 0 . Ton: a < 0,

Km a + 13 > O . 'Ecrtco ott TI <JUvciptTJ<rrJ f txet tfocrepec; pil;ec; 2

<JtO (a, 13), ttc; P1 < P2 < p3 < p4 . 'Etm, a1to to eeropT)µa tOU Rolle ' T) <Jl>VclPTIJ<ITI f ' ea txet tOUAclXt<JtOV tpeic; pil;ec; <JtO (a, 13), ttc; 11 1 E ( P P 2 ) ' 11 2 e (p2 , p3 ) Km 11 3 E (p3 , p4 ) . D µota, T) <Jl>VclPTIJ<ITI f" ea txet wut..cixtcrtov <ii>o pil;ec; crto (a, 13), µia crto <itcicrtTJµa ( 11P 11 2 ) Km µia crto (11 2 , 113 ) . Auto oµcoc; eivm P

f" J (a, 13) . Apa <JUvciptTJ<rrJ f txei t0 noM tpetc; pil;ec; crw <itcicrtTJµa (a, 13). a a E1tet<if] f(a)f ( ; 13 ) = a · ; 13 < 0, ano to eeropT)µa tou Bolzano yta TIJV f crto <itcicrtTJµa [ a, a ; 13 ] , TI

cit01to, ytati

, , , 'I

(

)

a + 13 Kut apa , , , , ElVm , µtKpOtEpT) , a +-13 . tOU :r a TIJc; f EtVat (Jt0 uta<JTIJµa a,-E1tEtat Ott µta tOU11.axtcrtOV p"::l s:

A<JKT) <JT) 2 . Mta cn>vO.pTl)Ot)

j f(x)dx 0

2 (1). Na mrol>E\;ETE OTl:

.!..2

1 f(x)

Auari. 'Exouµ£, yta Kciee e [O , l] : 1 � f(x) � 2 � -- ;?:

I 1 1I 'Etm txouµ£: J --dx ;?: - J dx

f(x)

El;ciUou, yta Kciee

f dvai optaf.1£vri Kat crovql]� ato &tO.aT1)11a [O, 1 ], ot TlflE� Tl)� avl]K01>V

G'TO OUl(JTl)Jla [ 1 , 2] Kat urxi>Et:

I 1 ;?: -1 (3). J --dx f(x) 2

::::>

j �x f(x) 0

.!. 2

(2).

•

x E [O, l] , txouµ£:

2 � 0 � -2- � 3 - f(x) . :Euve1troc;: [f(x) - l ] [f(x) - 2] � 0 � f 2 (x) - 3f(x) + 2 � 0 � f(x) - 3 + f (x) f (x)

I I 2 J-=--<lx � J 3dx -J f(x dx I

f( x )

(2).

)

}

2 J--=---<Ix � 3 - 2 I

f( x )

}

J--dx f( x) I

�

}

(4).

A1to ttc; (3) Km (4) faetm TI

A<JK'l<J'l 3. Na fJpEiTE Tt� cn>vap'rl]aE� f, ot o1to�� dvai optafll:vE� Kat cn>VEXE� ato &tO.(JTl)Jla [O, 1 ], I

J f (x)dx -3l + J f 2 (x 2 )dx

i:Tat cl>att va tcrxi>Et:

A6aT). 'Ecrtco ott µta <JUvciptTJ<rrJ f 1tAT)poi tic; <iocrµtvec; <JUvei]Kec;. E>ecopoi>µe tTJ <JUvciptTJcrri : g(x) x 2 , x e [0,1] . 'Exouµe: g(O)=O Km g(l)=l . Et..trxouµe ei>Ko/..a ott mx(>ouv ot u1toefoetc; tou eecopf]µatoc; avttKatcicrtacrri c;, yta ttc; <JUvapti]cretc; f Km g crto <itcicrtTJµa [O, 1 :Ewt, cri>µ<pcova µ£ to eeropT)µa auto (a1to to <iei>tepo µt/..oc; crto 1tproto) , txouµe: =

g(l)

J f(x)dx = J f(x)dx = J f (g(x) )g '(x) dx = J !( x2 ) 2xdx . I

Km E1tEt<ifi: .!..

g ( O)

J x 2 dx , ano TIJV <iocrµtvl) t<JOTIJta, txouµe: 0

I I 2 2 J /(x ) 2xdx = J x dx + J f2 (x2 ) dx I

•

J ( f(x2 ) - x )2 dx = 0 . 1

Tropa, <JUµnepcivouµ£ Ei>Ko/..a ott y1a Knee x e [0,1] tcrx(>et f(x 2 ) = x Km µ£tci ott f(x) = v'x (1troc;;). Avnatp6cp@;. D1tcoc; 13picrKoµm ei>Ko/..a T) <JUVclPTIJ<ITI auti] 1tAT)poi ttc; <iocrµtvec; <JUvei]Kec; Kat cipa eivm T) µova<itKi] /;T)toi>µEVT) EYKAEIAHI; B' 104 T.4172

To Bti1Ja Tou EuKAtie5n Mt9ooo� l:up1t/...l] promi� Tou TtTpayrovou

AtOv6CJ11 i; ruJ.vvapoi; ..;,. Ilupyoc;

H µt0oC>os tTJs «<n>µ7tA.ftprocrTJs wu tci:payrovou» civm yvrocr'tft, mpou avmpepctm t6cro crtrJv AA.yc�pa A' AuKciou 6cro Km crta Ma0riµattK6. 0cttKfts Km TcxvoA.oytKfts Katc'60uvcrTJs -B ' AuKciou. fta 1tap6.C>ctyµa ri c�icrrocrT) : yp6.cpctat C>tabOX.tK6.

x2 + y2 -4x + 6y + 12 = 0 (x2 -4x)+(y2 +6y) =-12 �(x2 -2·2x+22 )+(y2 +2·3y+32 ) =-12+22 +32 <=>(x-2)2 +(y+3)2 = 1

:EtTJ <n>VeX,ctCl 1tClpOUcrta�ouµc µta O'ctpa Cl1t0 1tClpClbctyµatCl O'tCl 01t0la <pCllVctClt 0 crT)µClVttKOs p6A.os au'tfts tTJs µc06C>ou. Ila p a�Etyµa 1 °

Na EKV 1) 1tapa<JTa<Jt) A 1UJ.Vi1J <11/

f ( x,y) = 3x2 + 14y2 - 12xy + 6x- 20y + ll

x, f (x,y) = (3x2 -12.xy+ 6x)+ 14/ -20y+ 11=3( x2 -4.xy+ 2x)+ 14y2 -20y+11 L1evwpo p1jµa: f ( x,y) = 3(x2 -4.xy + 2x)+ 14y2 -20y+11 = = 3[x2 -2(2y- l )x+(2y-1)2 ]-3(2y-1)2 +14y2 -20y+11=3(x-2y+ 1)2 + 2y2 -8y+8 Tpfro pljµa: f (x,y ) =3(x-2y+ l)2 +2(y2 -4y+4)<=> f (x,y)=3(x-2y+ l)2 +2(y-2 )2 llpdno pljµa: Ba�ouµc crc µta oµO.C>a t0us 6pous µc 1tap6.yovta to

<>riA.aC>ft

:Euµ1tA.riprovouµc tous 6pous tTJs 1tapev0ccrTJs rocrtc va <>riµtoupyficrouµc av6.1ttuyµa tctpayrovou

Ecpapµ6�ouµc tTJV iota µt0oC>o O'tOUs tpc� tcA.cutaious 6pous KClt ex.ouµc:

Ila p a�Etyµa 2°: Na 1Jp i:9o'6v 01 npayµaTtKt� A.t> ai:� Tt)� i:l;ia co<Jt)�

8x 2 + 44y 2 + 15z 2 - 32xy + 16xz - 44yz - 16x + 56y - 60z + 84 = 0

x' (8x2 -32.xy+16xz-16x)+44y2 +15z2 -44yz+56y-60z+84=0<=> <=> 8(x2 -4.xy+2xz-2x)+44y2 +15z2 -44yz+56y-60z+84=0<=> <=> 8[x2 -2x(2y-z+1)] +44y2 +15z2 -44yz+ 56y-60z+84 = 0 s[x2 -2x(2y-z+1)+ (2y-z+ 1)2 ]-8 (2y-z+ 1)2 +44y2 +15z2 -44yz + 56y-60z+84 = 0 <=> <=>8(x-2y+z-1 )2 +12y2 +7z2 -12yz+24y-44z+76=0 y 8 (x-2y+z-1 )2 +(12y2 - I 2yz+24y)+7z2 -44z+76=0 8 (x-2y + z-1 )2 +12(y2 -yz + 2y) + 7z2 -44z + 76 = 0 <=> 8 (x-2y+ z-1)' 12 [y' -2y(� -l)+ (� -1)' ] -12(� -lJ +7z' -44z + 76 0

A 1UJ.Vill<T1J :

Ba�ouµc O'tTJV iC>ta oµO.C>a tOUs 6pous µc 1tClp6.yovta to

01t0tc ex.ouµc:

Ecpapµ6�ouµc tTJV µe0oC>o cruµ1tA.ftprocrT)s tctpayrovou KClt ex.ouµc:

Ecpapµ6�ouµc tcbpa tTJV iC>ta btabtKacria yta tOUs 6pous µc 1tClp6.yovta to

KClt ex.ouµc:

Mc av6.A.orri cpyacria <o>

�

EYKAEIAH 8 '

104 T.4173

<o>

----�8(x-2y+z-1)2 +12(y- +1J +4z2 -32z+64=0�8(x-2y+z-1)2 +12(y- +1J +4(z2 -8z+16)=0� ! x -2y+z-1=0 �8(x-2y+z-1)2 +12(y- + 1J +4(z-4)2 =0 y- + 1=0 To

P1\µa TO'll Emtl.Eio11

-------

�

exouµe

�

A1t6 autftv mxipvouµt:

z-4=0

x=-1,y= l, z=4

Tiapa<>t1yµa 3 ° : Na ppt9t:i 11 d.ax1a'T1] T1J.l1\ T11� 1tapama<r11 �

x 2 + 6xy + lOy 2 - l2y - 4x

Ka9ro� Kat 01 T1µt� T(l)V x, y yia Tl� 01tOit� Clt>TO (J\)µpaivt:1.

A mi. VT1/<J'IF

Km 'teAtKa

J(x,y) =x2 + 6xy +10y-12y-4x = (x2 +6xy -4x) +10y -12y=[x2 + 2x(3y-2) + (3y-2)2 ] -(3y-2)2 +10y -12y= = (x+3y-2)2 -9y2 +12y-4+10y -12y= (x+3y-2)2 +y -4 y ( x+3y-2)2 + y2 ;?: 0 ( x+ 3y-2)2 + y2 -4;?: x2 +6xy+10y2 -12y-4x x + 3y -2 = 0 , , x 2 {y=O O

E1tet8t1

-4

ea eivm

s::

uT}r.auT} o-rav

Km y =

Ilott:� t:ivm a' at>Tft TflV 1tt:pi1tT(l)<r1} 01 T1µt� Trov x, y,

A1t6 TllV crxecrri

x +.

E lR .

A.pa 11

Ilapci<>t1yµa 4°: Na ppt:9t:i 11 t:l..ax1a'T1] T1µ1\ Tfl� 1tapama<r11 �

A mi.VTl/<llf :

tivm icrri µt -4 Km cruµpaivs1 6-rav

tA.ax1cr-r11 nµt1 Tll <; 1tapacr-racrri<; s::

ym Ka0e nµt1 't(l)V

z ;

x 2 + y2 + z2

oTav x + y + z = 1 .

y + z = 1 1taipvouµe 6n z = 1 - x - y t1toµevro<; 11 1tapacr-racrri

x2 + y2 +z2 =x2 + y2 +(1-x-y)2 =2x2 +2y2 +2xy-2x-2y+l 2x2 +2y2 +2xy -2x-2y+1=2 (x+-y 2-1-)2 + 23 (y- 31 )2 + 31 . , 2 (x+-y 2-1-)2 + 23 (y- 31 )2 ;?:0 x, ye 2 (x + Y 2- l )2 + �2 (Y .!.3 )2 + .!.3 ;?: .!.3 .!. 3 x + Y 2- I = O � x = .!.3 z = 1-x -y z = .!.3 . 1 1 y--=0 y=-3 3 AKoA.ou0rov-ra<;

T11V

i8m

8m81Kacria 11 -rsA.eu-raia 1tapacr-racrri ypacps-rm 8m8ox1Ka:

Etvm

Ka0s

cruµpaivs1 6-rav

{

. A.pa 11 sA.ax1crT11 nµt1 sivm icrri

Km a1t6 T11V

Ilapci<>t1yµa 5° : Na ppt:9o'6v 01 T1µt� Trov 1tapaµtTprov Trov 1:;1aroat:rov at>Tft 11 l..'6 <r11 . A 1Ca V Tl/<11f : •

•

{

µt:

ym Km

A.aµpavouµt 6n Km

a Kat fJ

yia T� 01toi£� TO a'6GT1]µa

x2 + 2y 2 - 2(a - 2/J) x - 4/Jy + a 2 + 6/3 2 - 4afJ 0 3x 2 + y 2 - 12x + 2 ( a + fJ ) y + a 2 + /3 2 + 2a/J + 1 2 0 =

tx.:1 J..'6 <r11 Km va

pp.:9t:i

01 s�1crrocrt1<; wu crucrtftµaw<; µt T11V µE0o8o cruµ1tA.t1procrri <; -rs-rpayrovou yivov-rm:

x2 -2(a-2,B)x+2y2 -4,By+a2 +6,82 -4a,B=(x-a+2,8)2 +2(y-,8)2 =0 3x2 + y2 -12x+2(a + ,B)y+a2 + ,82 +2a,8+12 = = (y2 + 2( a+ ,B) y + a2 + ,82 + 2a,8) + 3x2 -12x+12 = (y +a+ ,8)2 + 3( x-2 )2 = 0 EYKAEIAH B' 104 T.4174

��

, ,

E1toµEVco� to apx1Ko crucrtrJµa ctvm icroouvaµo µc to

icroouvaµo µc to

{(x-a+2/J)2 2 +2(y-/J)2 2 =0

To p�µa Tov EvKM:io�

��-

Km auto ctvm

(y+a+fJ) +3(x-2) =0 x-a+2/J=O x=2 x=2a-2 y-/J=O <=> y=/J <=> 2 1 ,a= 1, = - -1 . = x =y 2, y+a+/J=O a-2/J=2 a= l 2 2 x-2=0 2/J+a=O y=/J P=

Il a p ciottyµa 6 ° : Na ppt9Ei 11 µtyun11 Ttpl] TO'U lKUV01tOt0'6V 'TTIV E;foroaq

a1t6 61tou

yia 'TTIV onoia 'U1tapxo'Uv apt9poi x,y no'U

2x 2 + 2y2 + z 2 + xy + xz + yz

2[x' + ix. y;z +( y;z )' -( y ;zJ] +2y' +z' + ]" =4 <> 2(x+ y; z J- (y' + z:+z' ) +2y' +z' + ]" = 4 <> 6yz ) + 7 z2 =4�2 (x+y+z-)2 + g-15 [y2 +2y153z + (5z )2 - (5z )2] + g7 z2 =4<=> <=>2(x+-y+z4-)2 + S15 (y2 + 15 4 S y+z )2 + -15 [(y+ -z )2 - -z2 ] + -7 z2 = 4<=>2(x+ -y+z )2 + -15 (y+ -z )2 - -3 z2 + -7 z2 =4<=> <=>2(x+ -4 8 5 25 8 4 8 5 8

A 7UI VT11U1f ."

Il a p cioctyµa 7 ° : Ot apt9poi x, y,

Eivm TETotot cOO'TE

JlEytO"TTI Ttpl] Tll � napamaaq� 2x + y - z .

x2 + 3y 2 + z 2 2 =

•

Na ppE9Ei 11

2x + y-z =a:::=> z = 2x + y-a x2 + 3y2 + z2 = 2 x2 +3y2 +(2x+ y-a)2 =2 <::::> x2 +3y2 +4x2 + y2 +a2 +4xy-4at-2ay = 2<=> <> Sx' +4y' +a' +4xy-4xa-2ay 2 <> s[x' + 4 ( Y5-a ) x] + 4y' +a' -2ya = 2 <> <> 5 [x' +2x (2y � 2a ) + ( 2y � 2a J - ( 2y � 2a J ] +4y' +a' -2ya=2 <> [ 2 2 2 2y-2a 16 2ya a 2y-2a 16 a + (-a )2 - (-a )2 + -a2 =2 2 2 <::::> 5 (x+ 5 ) + 5 y --+-=2<=>5 x+ + y -2y ) ( 5 5 5 5 16 16 16 J 5 2) + 16 (y a )2 16 a2 + a2 = 2 <=> 5 (x + 2y-2a )2 + 16 (y a )2 + 15 2 = 2 <=> <=> 5 ( x + 2y-2a 5 S ( - 16 )2- 5 162 S 5 S - 16 80 ( 2 + -16 y- -a 2--a 15 2 0 , a2 --::; 32 ; O <=>- -32 ::;a:::; -32 , <=> 5 x+ 2y-2a ) 5 5 16 80 3 J¥3 J¥3 a = V{323 x= �8 V{323' y= -161 V{323 .

A1C<i.VT1f(llf." 'Ecrtco

Km avttKa0tcrtrovta� tr)V ttµil tou crtrJV

1taipvouµE

max

�

oriA.aori

1 1 S' r"

EYKAEIAH B' 104 T.4/75

. E1tOµEVCO�

� 0 �

« H Kapc5ui twv µa9riµanKwv Elvat Ta

E u KAci 6 n <:

n p o-.: c i vc 1 . . .

Ta KQl Ol Auaeu; Kal

npo�Mµa

KUplO<; Myoc; unap�ric; tOU

µa9ri µanKou Eivat va MvEl npopA� µata» . P. R. HALM OS

1a : r. K. TPIANTO:E - N. 0. ANTQNOilOYAO.E - 0. A. TZIQTZIO.E A.EKH.EH 268 (TEYXOY.E 98 ) crov2ro 1 = Na. a.1tooe1x0ei 6n yia. Ka0e a., p , y e R µe 4crov2 2ro - 1 3 (a. + p)(p + y)(y + a.) * 0 , icrx6ei ri a.vicr6tT}ta.: (�riµrr rpios Kaprcratlfis - Aypivio ) AY.EH (E>roµas TcraKac; - IIatpa. ) 1 1 1 9 1 � + Eivm: = B - ro, B, r = B + ro Km l a. + PI IP + rl + lr + a.I 2 · l a.I + IPI + l r l 1t (Hpatlfis EuayyeA.1v6s - fA.ucpaoa. ). 3B = 1t <=> B = A + B + f = 1t o1t6ts 3 AY.EH 1" ( fu.Opyos �eA.ricrta0ris K. IIa.tt1cria.) 1t 1t A = - ro, f = + ro . E1tstoi) E1tetoi) ( x + y + z )(_!_ + _!_ + _!_) � 9 , yta. Ka.0e ' ta. T}µitova. tmv 813 3 x y z 1tAa.crimv trov ymvirov tou tpiyrovou ABf sivm oia. x, y, z > 0 , yia. x = la. + Pl , y = IP + r l , z = lr + a.I ooxiKoi Opot a.pµovtKfis 1tpOOOoU, EXOUµe: 1 1 > 9 1 2 1 1 � -= -, UT}M..l.UT} .,... --,. .-. .. .,. -. ....,. ..,. .-..-.,. -+ + + la. + PI IP + rl lr + al l a. + PI + IP + rl + lr + al T}µ2B T}µ2A riµ2f Km Myro Tils tpiyroviKfis a.vicr6tT}ta.s: 1 1 2 9 9 � = la.I + IPI + IPI + l r l + l r l + l a. I 2(la.I + IPI + lrl > H icr6tT}ta. i<Jx6ei 6ta.v Ka.t µ6vov 6ta.v : la. + PI IP + rl = lr + a.I Ka.t a., p ,y oµocrri µoi, <>riA.a oi), 6ta.v a.=p=y. AY.EH 28 (:Eircriavri DA.ya 0€pµri E>ecr/viKTt s) 2� 1 1 <=> + --=,.....Me ecpa.pµoyt1 Tils a.vicr6tT}ta.s B - C - S exouµe: 3 �O'\.)v2ro - ri µ2ro �O'\.)v2ro - ri µ 2ro I2 1 1 1 I2 I2 = -- + -- + -- >crov2ro 1 2 .J3crov2m 2 J3 ---+ + <=> -<=> - = l a. + PI IP + rl lr + a.I la + PI IP + rl lr + a.1 2 3 3crov2 2m - riµ 2ro 3 4crov2 2ro - 1 9 (1 + 1 + 1)2 -> A6c:rq ECJTElA«V : Imawri s Av8pfis - A0fiva., Po l a. + PI + IP + rl + lr + a.I l a. + PI + IP + rl + lr + a. 1 �acpvri, IIavos f1awa.K61touA.os OOA<f>Os Mnopris E;arxsia, nawris Tcr61tsA.as Aµa.A1aoa., ftrop 9 9 ros TmroA.ris - Tpi1toA.ri, Krocrtas Nspourcros la.I + IPI + IPI + lrl + l r l + l a.I 2(1a.1 + IPI + l r l > rA.ucpaoa., �iowcrris riawapos - IIupyos, ftropyos A6c:rq tcrrt:1A«v: �iovl>crri s fiawapos IIupyos, AnocrtoA61touA.os MscroA.Ont, AvtffiVT}s Imawiftawris Tcr6m:A.m; - Aµa.A1aoa., �ri µfitPTJs Mavm 8ris - XoA.a.py6s" ftropros �sA.ricrta0ris Katro A61touA.os - Ka.tepivri, PoMA.q>os Mnopris - �acp Ilatt1ma., O µa8a npo PA.riµatmv 18u:onKou AuKsiou VT}, Avrrovris Imawi<>ris - XoA.a.py6s, ftropros AIlavayia Ilpoucrironcrcra - Aypiv10. 1tocrtoMnouA.os - MecroMni, �riµfitPTJs A.EKH.EH 270 (TEYXOY.E 100) Kapap6tas - AoucriKa Axa.ia.s, Xpi)crtos KupUl�tls Na osix0si Otl O"S Ka.0s tpiyrovo ABr Ka.t yta. Ka0s IIsipmas, fu.Opros TmroA.ris - Tpi1toA.ri, �fiµos v e N• i<Jx6si 6n: Ifo1ta.861touA.os - 'Eoscrcra., ftawris HA.16nouA.os A B CJUV cruv B f CJUV f A Ka.A.a.µata., E>mµas TcraKas - IIatpa, riawris :Eta 2 + 2 ;:::: 3 . 2 ----=2'--- + µmoyuivvri s - �poma, O µa8a npo PA.ri µfrr rov 181m f B A ri µ ri µ T} µ ttKou AuKsiou Ilavayia. Ilpoumcimcrcra - Aypivio. 2 2 2 A.EKH.EH 269 (TEYXOY.E 100) ( fU.Opyos NtKT}tO.KT} s - l:ritsia ) Tpiyrovou ABf 01 yroviss A,B,f sivm oia.0ox1K6t AY.EH 1" ( �iovl>crri s fuiwap0<; - Ilupyo� ) 6poi a.pi0µrinKfis 1tpo6oou µe oia.cpopa ro Km ta. A-B r A-B cruv -- 2cruv - cruv riµitova. trov a.vncrtoixrov Ot1tA.a.crirov ymvirov tou 2 2 = 2 = Eivm: a.pµoviKT) 1tp6o<io. Na. osix0si 6tt : r r r 2cruv 2 ri µ 2 TJ µ 2 Em SA.e

\;':

'1 - 1;': '

---

•

EYKAEIAHI: B ' 104 T.4/76

------- 0 E1ltdti0ttc; IlpoTdvtt . . . -------

A+B A-B 2 1'} µ -- cruv -- T}µA + T}µB a + 2 --�� 2 = --� = p 0T} µf T} µf y B-f f-A cruv -cruv -2 = y + a !:u2 = p+y µoiroi;, B A a p T}µT}µ2 2 venroi;, apKei va anooetx0Ei TI avtcronita a P a P+ ( + r + ( r r + ( Y + r ;;:: 3 . 2v . IIpayµatt, a p Y ano Tl'l yvroatit avtcronita tOU Cauchy exouµe: a+P r + a ;;:: �+r r < y r +( a r +( � ;;:: 3 · 3 ( a + � . � + r . r + a r = a y � a f af = 3( + � . � + y . y + ) ;;:: 3 · ( 8a�y ) = 3 · 2v onou a y � a�y fytve XPtl<JTt nii; : (a + � )(� + y)(y + a) ;;:: 8a �y , µe niv t<JOtT}ta va t<JX(>et µovo yta a=�=y. AYl:.H 28 (ftffi pyoc; AnomoA61tou/uJi; - MeaoA.Oyyt) . r A+B A-B Eivm: T}µ- = cruv-- Kat cruv -- > 0 . 2 2 2 A-B A B A B cruv -cruv - cruv - + T}µ-T}µ2 r =< 2 2 2 2r= ( A B A B f T}µ cruv -cruv- - T}µ-T}µ 2 2 2 2 2 A B l + E<p -E<p 2 2 =( A Br 1 - E<p-E<p2 2 B-r B r 1 + e<p-E<p cruv-2 2 f Kat 2 r =( oµoiroi;, ( A B r T}µ1 - E<p-E<p 2 2 2 r A r A cruv -1 + e<p -e<p f= f , onote T} anooetT}µ1 - E<p - E<p 2 2 2 Ktfo ypacpetm taoouvaµroi; ani µopcpfi: A B B r 1 + E<p- E<p 1 + E<p -E<p 2 2r+ 2 2 r +< ( A B B r 1 - E<p- E<p 1 - E<p - E<p 2 2 2 2 r A l + E<p-E<pr ;;:: 3 . 2 v (1) +( 1 - E<p-E<p 2 2

(

: i

(

; i

E>eropouµe

Tl'l

0 < u < 1 , Kat

�

f(e<p E<p

l+u f µe 1-u ( 1 ) naipvet ni µop<pfi: f(u) = (

cruvapni <JTt

�) + f(e<p � E<p �) + f(e<p � E<p �);;:: 3 · 2v

Enetofi ae Ka0e tpiyrovo ABf tcrx;Uet TI taonita: A B B r r A E<p-E<p- + E<p - E<p- + E<p-E<p- = 1 Kat Ot npo2 2 2 2 2 2 a0etfot eivat o/uJt 0ettKoi, npo<pavroi; tcrx;Uouv: B r A B Kat 0 < E<p-E<p- < 1, 0 < E<p-E<p- < 1, 2 2 2 2 r A 0 < e<p-e<p- < 1 onote T} teAf:utaia ypa<pfi nii; 2 2 anooetKtfoi; EXEt voT}µa µforo tT}i; f. Eni<JT}i;, eivm: 1+u . 1 , )v µe uEUtEpT} napayroyo f'(u) 2v( 1 - u (1 - u)2 1 l + ur f"(u) = 4v · -2 · (v + u) > O , nou 4 ·( (1 - u) 1 - u <JT}µaivet 6Tt 11 f dvm K1>pn\ G1'0 (0,1 ). Ano niv avtcrOTl'Jta tou JENSEN exouµe ani cruvexeta Ott: A B B r r A f ( E<p-E<p-) + f ( E<p-E<p-) + f(E<p-E<p-) ;;:: 2 2 2 2 2 2 A B B r Emi tG1'£tA.uv : ftropyoi; Tme0A:r1i; - TpinoA.T}, Po MA.cpoi; Mnoprii; - da<pVT}, lwaVVT}i; Avopi]i; - A0fi va, ftaVVTJi; Tcr6neA.ai; - AµaA.taoa, O µa8a npo _

PA.11 µ6.trov IOtrottKou AuKeiou Ilavayia Ilpoumro

ttcrcra - Aypivto AI:KH:EH

Av a, b, c, d

271 (TEYXOYl:. 100)

eivm 0ettKoi npayµattKoi apt0µoi tote va Oetx0ei Ott <JtO <JUvoA.o c t(l)V µtyaOtKcOV apt0�· ' a b c d µrov T} e..,taro<JT}: -- + -- + -- + -- = 0 z+a z+b z+c z+d µe z * -a, -b, -c, -d £xet µovo npuyp.«TtK� pi �£c; . (A eutEpT}i; TmA.taKoi; - faMtm) AYl:.H ( I16.voi; ftawaKonouA.oi; - E�<iPXEta) H oo0eiaa E�laCO<JT} eivat tpitou p a0µou µe npay µanKoui; cruvtEAE<Jtei; Kat apa OEXEtat touA.axt <JtOV µia pil;a npayµanKfi. Av oex0ouµe Ott T} E�i <JCO<JT} EXEt pil;a tov µtyaotKO apt0µ6 z0 e C - R tote 0a sxet pisa Kat tov crusurt1 tOU Zo E c - R . !:uvenroi;, tcrx;Uouv:

EYKAEIAHI: B' 104 T.4177

------- o Et1KM:io11� rrpo'Ttiv£i -- + --. + -- + -- = 0 a

b d c ( l ) KCll z0 + a z0 + b z0 + c z0 + d a c b d + =--- + + = 0 (2). z0 + a z0 + b z0 + c z0 + d Me acpaipeITTJ T11 i; (1) an:o T11V (2) n:aipvouµe : 1 1 1 1 a (=-- - -. ) + b(=-- - -- ) + z0 + a. z0 + a z0 + b z0 + b 1 1 1 1 +C(=--- - --) + d(=-- - --) = 0 � z0 + c z0 + c z0 + d z0 + d =---

=---

d zo-Zo"o00 a + b + c )=0 � l zo + di 2 l zo + al 2 l zo + b i 2 l zo + cl 2 d =0 (3) + zl o + d l 2 Aton:o, mpou ot n:pocr0etfot tou n:ponou µEf..oui; ei vm 0ettKOi. Apa TI e�icrcocrri OBXe'tClt µovo pi�ei; n:parµattK€i;. Ai>Cfll t<rrt:iA.uv: Iwawrii; Av&pfi i; - A0t1va, Po MA.<poi; MnopT1 i; - 1'.acpv..,, ew µai; Tcr aK ai; - Ila tpa, 1'.wwcrrii; f tawapoi; - Ilupyoi;, f tc.Oproi; Ano crrnA6rrouA_oi; - MecroMrri, ftc.Opyoi; 1'.ef...ri crta8rii; Kfrrco Ila'ttl ma, O µa&a rrpoPA.T1 µcitwv l&twnKou AuKeiou Tiavayia Tipoumc.Oncrcra - Arpivto. +

AI:KHI:H 272 (TEYXO� 100 )

• • .

--------

tp = �t(t - a)(t - P)('r - r) � tp 2 = (t - a)(t - P)(t - r) � tp 2 = t3 - (a + P + r)t2 + (ap + Pr + ra)t - aPr = t 3 - 2t3 + (ap + Pr + ra)t -4Rpt ·� p 2 = -t2 + (ap + py + ra) - 4Rp � aP + Pr + ra = t 2 + p 2 + 4Rp (4) H crxecrri (1) pacret tCOV (2),(3),(4) rpacpetm: x 3 - 2tx 2 + (p 2 + t2 + 4Rp)x - 4Rpt = O (5). Apa, TI (5) exet pi�ei; toui; apt0µoui; a,p,r. Av te0ei f(x) = x 3 - 2tx 2 + (p 2 + t 2 + 4Rp)x - 4Rpt tote TI f exei tpeii; omcpopettK€i; µeta�u toui; pi�ei; on:ote cri>µcpwva µe to 0effipT1µa tou Rolle TI n:aparcoroi; T11 i; f'(x) = 3x 2 - 4tx + p2 + t 2 + 4Rp EXet aKptPc.Oi; ouo avtcrsi; pi�ei; µe an:ot€A.ecrµa TI OtaKpivoucra T11 i; Va eiVCll 0etttj. 1'.T1AaOtl eivm: 1'. > 0 � 16t 2 - 12(p 2 + t 2 + 4Rp > 0 � 4t2 - 3(p 2 + t2 + 4Rp > 0 � t 2 > 3p(p + 4R) AYI.H 28 ( ftciwT1 i; TcroneA.ai; - AµaA.taoa ) A

An:o tov voµo tcov Tlµttovcov crto tpirwvo ABf €A A xouµe: a = 2RTlµA � 4RT1µ CTUV = a (1). An:o 2 2

Na an:ooeix0ei ott m µt1Kl1 a,p, r twv n:A.euprov tpirrovou ABf µe a < p < r eivm p�ei; T11 i; s�icrwcrri i; : x3 -2tx2 +(r +p2 +4Rp)x -4tRp =O (1) , on:ou to tpircovo AM n:aipvouµe: crcpco = M 11'. , OTIA t TI Tlµtn:epiµetpoi; tou tpirrovou Km p, R ot aKtivei; A tcov eyyeypaµµevou, n:eptrerpaµµEvou Ki>tlcov tou CTUV A t-a t-a 2 = -tpirrovou avttcrtoixcoi;. :ET11 CTUvexsia, va an:o&et- crcpMe (2). = -- � -A p 2 p 2 x0ei ott t<JXl)et TI avmoT11 ta: t > 3p( 4 R + p) . TIµ 2 (Avi:c.Ovri i; Iwawi&rii; - XoA.aproi;). n:oA/macrµo Kata µ€A.TI tcov (1),(2) n:aipvouµe: A�H 1 8 (O µa&a Tipo P"-Tl µai:wv I&iwnKou Au A a(t - a) A a(t - a) Ksiou Tiavayia Tipoumc.Oncrcra - Arpivio.) � CTUV 2 = 4RCTUv 2 = (3) 2 4Rp 2 p H e�fococrri µe pi�ei; toui; apt0µoui; a,P,r eivm (x - a)(x - P)(x - r) = 0 KCll moouvaµei µe T11V ap A evro µe OtaipeITTJ Kata µEA.TI: Tlµ 2 = (4) x 3 - (a + P + r)x2 + (aP + Pr "+ ra)x - aPr = O (1). 2 4R(t - a) (2). Eivm a + p + r = 2t A A En:etOtl TIµ 2 + CTUV 2 = 1 an:o tti; (3 ),(4) exouµe: aPr 2 2 = pt � aPr = 4Rpt E = pt � (3). 4R ap a(t - a) � ap 2 + a(t - a) 2 = l =l + E = �t(t - a)(t - P)(t - r) � 4Rp(t - a) 4R(t - a) 4Rp � ap 2 + a(t - a) 2 = 4Rp(t - a) � ap 2 + at2 EYKAEIAH� B ' 104 'T.4/78

-------

o E1>Kl..tio11 <; IlpoTtivti

-2o.2t + o.3 = 4Rpt - 4Rpo. <=> a.no o.3 - 2to.2 + (t2 + p2 + 4Rp)o. - 4Rpt = 0 (5) TI'IV (5) cruvayouµt Ott 0 a eivm pll;o. TI'I� 3 x - 2tx2 + ( t2 + p2 + 4Rp) x - 4tRp = 0 . Oµoico� o.noC>etKV'Uetm ott Ko.t ot apt0µoi p,y eivm pi�e<; TI'I<; e�iCJCOOT)� autTi�. To C>f:Utepo CJKEAo� TI'I� o.noC>et�rt� eivm ioto µt auto nou eµq>o.vi�etm CJTI'IV 1 TJ AUOTJ. Avent ttn£lMiv: atovucrri i; ftawapoi; - Ilupyoi;, ftwpyoi; AnomoA6nouA-oi; - MecroA.Oyyt, 0ro µai; TcraKai; Ilatpo., Iroawrii; Av8pi] i; - A0ftvo., rra voi; ftawaKonouA-oi; E�arxeio., Kwcrrni; Ncpou rnoi; fA.uq>aoa, Po86A-<poi; Mn6prii; - .:laq>VTJ, ftwpyoi; �eA-rima8rii; - Katco IIo.tTicna. AI:KHI:H 273 (TEYXOYI: 100) .:livetm tcro<JKeM� op0oyrovto tpiycovo ABf ( A = 90° ) U'lfOU� AM = 2a Ko.t 0 to µfoov tou AM 0 K'61<Ao� Oto.µ&tpou AM teµvet TI'IV Of crto OT)µtio .:l Ko.t rt B.:l TI'IV OM crto OT)µeio E. Na C>et, OE = <l> = 1 + .J5 x0 et, ott: 2 EM ( ftwpyoi; T p1avwi; - A0ftva ). -

AYI:H (

Iroawri i; Av8p i] i;

- A0i}vo. )

Aµo.A.1aoo.,

E�apxeio.,

rA.ucpa8o.,

rravoi;

O µa8a IlpoPA-ri µa

trov 181ronKou AuKtiou Ilavayia Ilpoumwncrcra

Aypivto.

IIPOTEINOMENA 0EMATA

298.Av a,p,y eivm to. µftKrt tcov nA.wpwv tptyrovou ABf Ko.t R, p ot O.Ktive� tou neptyeypaµµtvou, ey yeypaµµtvou K'Utlcov tou tptyrovou ABr avncrtoi xcoi;, tote VO. o.no8eix0ei Ott tCJXl}et : R p+y y+a a+p + + �3 (1 ). a. + p - y p + y - a y + a - p p ( ftci:>pyoi; AnocrtoA-6nouA-oi; - MecroA.Oyyt ) 299. Av yto. tou<; 0ettKOU<; npayµo.ttKOU<; o.pt0� a,p,y mx;Uet on: a. + p + y = 3 , tote vo. C>etx0ci 6n: 5 - o. t3 - 5 - P pJ - 5 - y 1i ) ( ( ( ) � S (l) . 3 -P 3-y 3 - o. ( ftwpyoi;

AnomoA6nouA-oi; - M�

300 . .:livetm tpie8po<; yrovia

)OXYZ µt £()puc&; r&

vie<; YOZ = a, ZOX = p,XOY = y JCm cmivavn npo<; aut&<; 8t&opoui; A,B,r avrurroqm;. Na wto oeix0d Otl tcrx;Uouv Ot no.pO.KQt(l) \(J� flµo. 11µP 11µy I) = = (1) /\

{

Kwcrrni; Ncpoutcroi;

ftawaK6nouA.oi;

11 µA

....

-------

• • •

11 µB

11 µ[

cruvo. = cruvpcruvy + 11 µ�11µycruvA (2) II) cruvp = cruvycruva + 11µrriµacruvB cruvy = cruvacruvp + riµariµPcruvf (rtwpyoi; Tptavtoi; - A0ftva) 301 . An6 TI'IV Kopucpft M tpiyrovou KAM cpepouµe to U'lfO<; A0 µi}Kou<; h , 'tTl C>taµecro MP Ko.t 'tTl C>t xotoµo MN To OT)µeio N dvm to µfoov tou 0P. H anocrtaOT) TI'I<; Kopucpi}<; M a.no to op0oKevtpo H tou tptyrovou autou eivm d. Na unoA.oym0ei to U'lfO<; TI'I<; <>ixotoµou MN ( ftwpyoi; T ptavtoi; A0ftva ) 302. Av 01. apt0µoi a,p,y µt a < p < y dvm pil;e<; TI'I<; e�icrroOTJ<; x 3 3x + 1 = 0 tote va unoA.oytcr0ei 11 ttµf} tOU a0poicrµato<; S = 0. + � + J_ . p 'Y a ( �towcrrii; ftawapoi; - Ilupyo<;)

Ano to tpiycovo OMr Ko.t to 8£<bp1)JUl Tot> M£V£Mot> µt foo.teµvoucro. TI'IV BE.:l exouµe: BM . .:lf . EO Bf . ao = 1 <=> EO = Bf .:10 EM EM BM .:lf EO .:10 EO 2 � <=> EM = 2 .:lf <=> EM = .:lf (1) Ano to IBto tpiycovo no.ipvouµt on: rM2 = Or2 - 0M2 � 4o.2 = (a. + af)2 - a.2 � 40.2 = .:lr2 + 20..:lr � .:lr2 + 20..:lr - 4a.2 = o � .:lf = a.(JS - 1) (2) 303. l:e KUpto tetpanA.wpo ABra to a0poicrµo. a.no TI'IV ( 1) MryCO TI'I� (2) EXOUµt teAtKa: tcov tetpo.yrovcov tcov nA.wprov Ko.t tcov oio.ycovirov ../5 + 1 E0 a. tou eivm L. Na 5eix0ei on yta to eµpo.Mv tou E = =2 EM o.(.J5 - 1) 2 tcrx;Uet 11 crx&OTJ: 8E � L . A'6crtt ttn£lMiv: ru:Opyoi; TmwA-ri i; TpinoA.ri, at ( �towcrrii; ftawapoi; - Ilupyoi;). owcrrii; ruiwapoi; - Ilupyo�. 0co µai; TcraKai; Ilatpo., rtwpyoi; AnocrtoA01tOUAOi; - MecroA6yyt, EYXOMAI:TE KAAEI: AIAKOIIEI: ! ! ! . Po86A-<poi; Mn6prii; .:laq>VTJ, ftawri i; Tcr6m::A-ai; .

EYKAEIAH� B ' 104 T.4179

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