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apa 0 J.ttKpon:po<; ea elVat Jl0Vcl06<; OEKa J.telOV eva apteJ.t6.

0 -

6 aqa €A0:aawv ea'tat M t

"'Ca

7tOL€LV

M.,.;

11\

I I I I I

�

A)J..a 'tO ytVOJ.t6VO amrov 6ivat iao Jl6 JlOVcl06<; 6Ka't6 J.telOV Eva 't6'tpayrovo

!1Ya

............................................................................................................ - ....................... __..,.. _...................

-rav-rataa M.,c; 0

Kai. ytV€'taL 6 c MP 0

Ea'tat aqa 6 flEV fl€Ll:wv M tP, -u•u--�--,.,..__,.._,.,__ u•u•-u--•u-�•-u--•-uo-""u'"'"'"'"____,.,,. ___,.,__

6 b£ tAaaawv M 11 Kal 7tOLOUOLV 'tCl rij� 1tQO'tlXU€W�.

x +"'= o+1 o +1 o .....o = 20 KatX-'If = 2o (2) ---------��-----

-�.............-......... ....,...............".�""""''""""'-"'"___ ........._....., ............. ............... .................... -·--··..---·-------------·---------

...... .. . ... ........

·-·-·

ai\i\' 6 un' au-rwv €an

-------

... ............ ..... ..

"'= 10 - o

'tO ytVOJ.l6VO 'tOU<;tivat Jl0Vcl06<; INevtlV'ta E�t.

i\om6v €an Kai. -rov un au-rwv 0

Kat (1tapa)J.lEv6t 'tO Jl6V cl9potaJ.tcl 'tOU<; Jl0Vcl06<; eiKom , 11 06 ota<popa 'tou<; Mo apt9J..tOi.

Kal flEVH 'tO �£v aUV9€fla MK, 0

tl b£ U7t€QOXi] c p

-····-------·-·-·-·····

X .'If= 96 ········· .. ····· .. ·······------·..···----

····--···-·-·---·-·--·-·····

x .'If= (o+10 )(lO-o) = 100-02 (3)

Kat dvat au'ta iaa Jl6 Jl0Vcl06<; evevilV'ta E�t.

'"'"'"'"'"'"..,_"'"""""""' ... .,._,.,,.�,,. ... ,.,..,,.,,.,.,,.,..,.. ... ....,,.,,.,.,._,..,....��w•�•w�---

...... ....

.. . .......

····· · · ·· · · · · ·· ····· · · ·· ·· ········· ···· · · · · · · · · · · ·-·-·····

100-02= 96 (4) !

Kat o apt9J.t6<; 6ivat ouo Jl0Vcl06<;. 'Apa o J.t6V J..1£Yal.:6't6po<; ea eivat !!;0Vcl06S 00006Ka o 06 J!tKpO't6pO<; J.l0Vcl06<; OK'tOO Kat 61taA.118e6ouv 'f11V 1tpO'ta<J11 .

apa

o= 2

! x= o+10 = 2+10 = 1 2

� ............................................................................

"'= 10 -2= 8 x+\jf= 1 2+8=20 X \jf= 1 2·8 = 96

(5)

EnlGf1JI.avc:n:� : • (1), (2): EA£r xt'tat 11 taoouvaJ.tia a'to 1tp6PA.mta, J.te'ta 'ta 't6xvaaJ.ta'ta Kat n<; 1tp�6t<; . • (3): eerop6i o6ooJ.tEvll 't1l r vro<lll 'tll<; 'tau't6'f11'ta<; (a+p)(a-P)= a2- P2 (&a<popa 'te'tpayrovrov) • (4): H OtroWJ.ttKit t�iaro<J11 x2= a Kat 11 eenKft 'tll<; P11Ttl AU<J11 . • (5): Kav6t 61taA.t19eu<J11 'tOU 1tpopA.fl J.ta'to<;. • I:'f11V 6K<f>OOV11<J11 'tCOV 1tpoPA.11J.tcl'trov oev XP11atJ.t01totei (Jl)JlpoA.taJ.tOU<; , evro an<; A.ua6t<; 'tOU 6J.t<pavi/;oV'tat J.t6 <p6tOro.

AYI:H TOY IIPOBAHMATOI: I:TH fENIKH MOPcJ)H TOY: � !E '10 �Ao:t�

'1tn'"

{

3104J�_'Ea'tro x-'1'= 2o , x>'lf, 'tO't6

X,+lji=Q

�'I' = � ,

a a apa x = X+\jf+X \jf=X+\jf+O = +O Kat 'If= X+\jf-X \jf=X+\jf-0= -0 2 2 2 2 2 2 2 2 . 2 o =p � IIapa'f11pouJ.te x+'l'=a Kat x-'1'= 2o , x·'l' = p � +o -p = K2• 'Apa 0= K ' 01t0't6X = o+

{

X+lji=Q X·IJI=P

�

{

lji=Q-X X·IJI=P

�

{

Kat "'=

lji=Q-X x·(- x+a)=p

-o=

)

( �)

- K. 2 2 2 2 Ot nJ,lt<; 'tCOV x, 61taA.118e6ouv n<; 1tpofuto9ea6t<; 'tOU 1tpopA.iJJ.ta'to<;. ME TH l:YNH8H AYl:H l:HMEPA: (Ava/;;11'tOUJ.l6 n<; 96nKe<; P11'tE<; pi/;;6<;)

'I'

= K+

(% )(�

�

{

lji=Q-X xLax+P=0(2)

EYKAEI� B ' 89 T.l/3

····· ·------

------

B ' A.1.lKdou Ken -ru «nl.uopu-rlKm> npopA.qpu-ru TO\l At.Ocpuvro\l

H (2) eivat e�iO'OlaTJ omnepou Pa9J.lOU J.l£ OtaKpivoucra ll= a2 -4P = 4[ a± .J4K 2

, Kat A.ucre� t� x 1,2 = .s:: • x > E1tewn 'I

•p

2 a

a± 2K = a - ±K. 2 2

( �)

- p] = 4K2 ' K>O.

a a , , n \11 = a - x s:: • utvet \11 = a- - K = - -K . tote x = - K ' onote ' 2 'I 'Y 'Y 2 2

To ttpdflh/pa 28 (q)

ftO

BI/JUo I;rOJv aplfJJI'Iftl«iJV TOV .d�v dwu:

KELJ.lEVO GTl)V EAAl)VtKt]

Keipevo GTl) NEOEAAl)VtKt]

TtTOlOU<; cOO'T€ _Kal. l) m)v8�au; atJTWV .!l.�e9��e�ft ��<; K�t 'W a8pOlO'J.la TCDV 'tc'tpaycOVCDV Kal. � m)v8wu::; TWV an' (Xl)'(WV -rou<; va eivat icra _'(_E_�Q0_Y�'!W'!_��_Li.i_. J.l€ 8o8tv•e<; apt8J.lOD<;. oo8£v1:ac; aQt8f..lOUc;. --·-- _

··-·--

-···

______________ _

-�!:.1...�-rL.. Jipt?.t�t, en�nA.t�N 'tO 8mA.amo TOU a8poicrJ.laW<; 'rOUe; bl.c; an' (Xt)'(WV TCDV T€TpaycOVCDV WU<; Va TETQaywvouc; 'WV ana civat J.lcyaA.U'tepo an6 'tO auvaflcpOTEQOV at}'(WV •e•paycovo -rou a8poicrJ.la-ro<; TETQaywvov '6mQ£xav 'tO Ka-ra •e-rpaycovo apt8J.lo'6. .. ............ . ........................................ TETQaywv4-J. •e-rpaycovo apt8J.l6. ----· · ··----------

Ma91')J.lUTtKt] -Gllppo.AtKt] ypaqn'J Na �pe8o'6v ot apt8J.loi x, \jl.. ., -rtwwt O:m-re :

Kat tcr-rco, 6n o J.lcyaA.u-repo<; eivat tva<;

·························---·····--···---··--·-········--·--·········· ·

Kat o J.llKpO'tepo<; J.lOVU8c<; 8tKa J.lclOV tva apt8J.l6.

�

' '

. ...... .......... ................................... __

cruv8ftKTJ (nepwptcrJ.l6)

2y - a = K2 (= -rtA.ew •e-rpaycovo)

rsviKSUO'Il

i a ! X = 8+ -

11 8ta<popa wu <; 8uo apt9J.loi.

�

X +\jl = 20

...................... ............

··· ······- --·------···-------

__ _ ___,_

. .

-.......................

x- o/ = 28 ( 1)

2 2 x + \jl = 208

......................-... ..................... .. ...............................___

_________________ _______

1 o/ - 2 - 8 _

-J...... I X +\jl = a

-----------------··········-·-·--·-·--------·····>·•-· ·�·--··-····---------------- ·-�-··-·-·----·----�-·-···-·-�--·---·-·--�·-·-----�-----·-

Eivat, AOl1tOV 'tO a8pOtcrJ.la "CCDV TcTpaycbvcov "COD<; J.lOVa8e<; 8taK6crte<; OK'tcO ..__ ... ..... .. .... .......... ........... ..... .

.-..................______________

\jl = 10 - 8

.......... _______ .....

To a8pOlO'J.lU TOU<; J.ltvct 1tUAl eiKOO'l

�

_ _____________ __________________ .

,� i

x- o/

28 ( 1)

-------��--�------·-----··--·

2 \j/2 x+ =y

.___ ______ ,_,.,,.,.......J....___..,_______

EYKAEIAm B ' 89 -r.l/4

Ta a, y tKavonmouv 'tTJ

.......... . .......______ ....... ....... .

·-··· .•

6nou a, y yvcomoi apt8J.loi

X = 8+ 10

apt8J.l6<; (C� ) cruv J.lOva8e<; 8tKa ( Mt) 0

� -·

{

Ms TOuc; apt9J..LODc; 'WU D.tO(j)UV"COU

-------

_______

------

B ' I..1>Kdo1> Kat 'ta «nl..aopa'ttKa» npoiJI..qpa'ta 'tOO At.O<paV'too

AJJ..iJ.

'tO a8pOUJf.la 't<OV 'te'tpayrovrov a1>-rrov, Kavet QUO -r�::-rpayrova Kat f.lOVUOe� otaKo<ne� (� � M cr )

AJJ.. a x2+ � =(o+ 1 Oi+(1 0-oi = 252 + 2·100 (2)

Au-ra �::ivat icra f.lE otaKoma oK-rro Kat apt8f.lo� yive'tat f.lOVUOe� ouo (CM � ) (ini. 'tac; unocr'taanc;), eivat 0 f.le\' f.lE'YaAU'tepo� f.lOVUOe� orooeKa ( M t�)

252 + 2·100 = 208

0 -

apa o = 2

-------

Eivat, x = o+10 = 12

K a E,tvat: x = -+2 2

Ent011 f.lUVcret� : (1): EAkyxe'tat 11 tcroouvaf.lia cr-ro 1tpO�A11f.la, f.le'ta -ra -r�::xvacrf.la'ta Kat 'tt� 1tpa�e� . (2): E>eropei OeooJ.Iivrl 'tTJ "(VOOOTI 'tTJ� 'taU'tO'tTJ'ta� : (a+�i + (a- �i = 2a2+2�2 1tOU OTif.laivet on Ka-ra 'tTJV e1tOxrl -rou , 'tTJV XP11crtf.l01totoucrav xrop� totai-rep11 ava<popa. (3): Kav�:: t e1ta.A.ft8eu011 -rou 1tpo�A.ftf.la-roc;. Xp11crtf.l01totei 'tTJV iota alla'Yft f.lE'ta�A.11'tft<; f.lE -ro 1tpo�A.11f.la 27, Kat 111rapoucria011 'tTJ� AUOTI<; eivat 1tot0 crUV't0f.l11, xropic; va xavet a1tO 'tTJV aKpi�elU 'tTJ� ( 0 f.laSTJ'tft�- avayvrocr'tTJ� £xet TJ011 e�otKet<08ei f.lE 'tO 8£f.la). • •

•

'J1=1l-X

x2 +(-x+af=y <=>

{

'J1=1l-X

{2) eivat e�icrro011_ oeu-repou �a8f.lOU f.lE otaKpivoucra �= 4a2-8(a2-y) = 4(2y-a2 ) = 4K2 , K>O. , X = -a +K , a K E1tetvn X > ,., 'tO'te 2 2 ' 01tO'tE ,., = 2 2 H

s:•

{

l" A'l'>cnt:

- -

{

( yta a = 20 Kat y = 208, o1tro� o �to<pav-ro�)

X+1j1=20

x 2 +1J12 =208

<=>

X+ljl=20

{x+IJif-2xljl=208

<=>

X+ljl=20

X·1J1=96

{

Au-ro OTif.laivet on �::mMyov-ra� -rou apt8f.lOU� au-rou� ft8�::A.e va Oe�et 'tTJV cruva<peta (oumamtKa 'tTJV X·IJI =96 tcrOOUVaf.lia ) 't<OV QUO 1tpO�A11f.lU't<OV ( Kat f.lE 'tO crUcr'tTJf.la x2+1J12=208 ), O'taV f.lE'ta9) 't<OV a, �' 'Y lcrxUet 11 crx£011 a2 = 2�+y.

EYKAEIAHl: B ' 89 't.l/5

KEif.lEVO CJ'TllV EA.A.l1V1Kll

.9!t.�S .

KEif.lEVo CJ'Tll NEOEAAllV1Kll

... . .... ... � umQoxi] av'rwv KaL 6 ...

. ..

..... . . .

............ ..... ....

.....

...........

noMar(t\acnaCJf-lOc;

'tE'totou� ci:l��E

Ma9llf.l«'T1Kll -CJ1Jf.1PoA.tK'Il ypa<p'll

11 ota<popa wu� Kat ....

o no))..,anA-acrtacrJ.L6� LOU�

n•«mn<n.-<mw.n.. .-�n-.-m•n-•n-"'""""""'""'''''""'""""'""'.-�Hmm<n<nm""nmnnHn<n<n«""m" "' '"'

"''" "

""'"'""'"'"''""''"

'"'"'""',., "'" -._,.,.,.,"'"""" ,

" ""

, ,

''"'"""""" "

va EtVat icra J.LE 008Ev't£� --...........-....... ... ... .. ,, �.P..��.l':.<?.��· . . . . . . . . . . . . . . . . . . . . . . . . . . .

nou� oo9£vmc; cXQL8f-lOUc;.

6no1J a, B wcocrroi apt8uoi

fl.ci -�iL . .. .. .. . ... ----....................................!:!P..��.��.? �.�.��.��?Y. .. . . . . . . . . . .. . Ta a, '[QV 'r£'!:QcXKLc; un' aV'rWV f-lE'l:Gt LO 'tE'tpa1tAUCJto LOU .

--····"'"'"""""""""____ ............. . ...... .

................. . ....___

. .

....

.........

't'OU ana 'rfjc; U71EQOXfic; aV'rWV ....

. ..

....... .

.... . .... . .. .. .. . ...

....

ytVOJ.LEVOU LOU� CJUV LO

............................ .. ................................. �?�P�:YC9Y9..�<?:t!.��P()!9'Jle1:1:<?S.....

noLEiv 'r£'rQaywvov.

i£crn Of. Kat LOlJLO 7tAacrJ.LanK6V

........,_,_____________"'""""-"""""___________ ___""""""""

va Eivat icra J.LE 'tE'tpciycovo

. . .�.P.�.�. Jl2.:

r:t 1-'

tKaV01t0lOUV •'11

(

cruv8 Kll nEptoptcrJ.L6

)

.................................................................................................. ..

civat Kat aU'tO 7tAacrJ.LaLtKO.

A vnq rov npopJ.Ij}laro�, lJt''l yevuaj rov JIOpffJij, JIB rq }180o�o rov dto(/Javrov:

{

X-'Jf= a( = 4)

X·'Jf= P( = 96)

Ti eptOptcrfJ.O£: 4P+a2 = K2• As eeropijcrOUfJ.E X+ "' =2o . TotE fJ.E'YaAUtEpos eivat 0 X= o+ 2a (= o+2) Kat 0 fJ.tKpon:pos eivat 0 "' = o- 2a ( = o-2 ) (tO clepOt<JfJ.cl tOU eivat 2o Kat '11 Ota<popa tOU£ X- 'Jf 1tapaf.l.Evet a ( = 4) ). = p. Ti pbtet x·"' = p ( = 96 }, aUa x·'l' = (o+ � )(o- = o2 Eivat 1taAt x = a2 + K2 ( = 12) Kat 'I' = a2 - K2 ( = 8 ), 1tou E1taAll9e'6ouv to 1tpoPAllfJ.a.

(�J

�)

l:xoA.aa: a) :Eto 1tpoPAllfJ.« 27 eeropei X-

"' = 2o Kat U1tO/..oyii;et o=2 ' apa X- '1'=4 1t0U crtO 1tpoPAllfJ.a 30 eivat l1 fJ.la e�icrro<Jll tou crucr'Ti]fJ.«tO£. :Eto 1tpoPAllf.1a 30 9eropei X + '1' =2o Kat ppicrKet o= 1 0� apa X +'Jf =20 1tOU crto 1tpoPA.llfJ.a 27 eivat u1to9e<Jll . r----------------------- ��---------1

r Ot apt9fJ.oi: Br = a = 5 J.tOV. : ........... , ............ , .......... , .... � p fA = = 4 f.lOV. 20 = 5 ·22 ' : 96 = 3·26 ' : AB = y = 3J.WV. ....... � BM = � = .Jl3 f.lOV. : 208 = 13·24 eivat KatacrKeuamfJ.Ot. - :- - T --.-�::---:-:Eto orotepo PtPA.io Kat crLl1 AU<Jll tou 1tpopA.i]fJ.atos 9 , o dto<pavtos ava<pepet 13 = 22 + 32 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -· E1ti�s crto 1tpoPA.llfJ.a 30 tou iOtou PtPA.iou, ava<pepet 'TllV tauto'Tllta a2+P2±2ap = (a ± Pi Kat �ava on 3 + 22 = 13 (yta a = 3 ,p = 2). •

3 e

y=3

•

I I

.- 1

�-

EYKAEIAIIl:. B' 89 T.l/6

1' I I I I I I I r

---•

B' l.'VKtiou Kat Ta «nl.aopaTtKa» npo1Jl.1lpaTa Too Aw«pavto'V

{X+1J1 = 20 I:to 1tp6PA.TJJ.l<1 27 X = e�icrroO'T) X+'JI=20 1t<lp1<rtclvel eu9eia ' evro e�icrroO'T) X'JI = 1tap1crtave1 U1tepPoA.ft. E1toJ.ttvroc; 01 cruV't£tayJJive<; trov KOtvrov touc; O'T)J.tCirov, eivm ------

y) rt(J)JIE1'j)\KU :

•

·IJI

ot A.Ucretc; tou crucr'tftJ.tatoc;.

AvaA.Oyroc; to 1tp6PA.TJJ.l<1 30,

·ljl

' T)

'�'

4 96

{

"""PPol.� X'If� 96 Kat to otlmtJIU' X +'JI = 20 � 'tOV JC6KJ.o X2 +� =208.

------

, eKcppa�et ta Ko1va O'TJJ.tCia 't11<; eu9£iac; x-'1' = 4 J.lC 'tTJV X+\jl=20 2 + 2 =208 ta O'T)J.!Ct<11tOU tEJ.lVetat TJ eu9eia 'I'

:E'to crxfiJ.la cpaivetat 11 eu9eia x+'l' = a = 20, 11 '1>1tep�ol.:r'] X'l' = � = 96, o K(ncl.oc; x +� = y =208 Kat 11 eu9eia x 'I' = 2o = 4. AxetKo�ovtat , 011A.aoi) Kat 'ta 'tpia npo�A.iJJ.la'ta 'tou J.!BYOAoU J.la�J.lanKoo (Ka9roc; 'to 1tp6�A.11J.la 1).

- - · · ·

!· · · · · · · · · · ···: · · · · · · · · · . .. . . .. .. . . .

..... � .. . .... . .... · : . . . . . . . . . . . . .. . . . .. .. .: :

;-20

--------------------------------------------------------------------------

Ot KrovtKec; 'toJ.ttc; 'teJ.lVOvtat <na OTJJ.teia A(1 2, 8), B(8, 1 2) 1tou eivat Kat ot A.ucretc; 'trov avn<noixrov 1tpo�A.mta'trov. 0 J.lB'ta<JXI1J.lancrJ.l6c; x-'1' = 2o Kat crnc; ouo 1tpotacretc; oivet 0 = 2 ' 01tO'te 1tpOKU1ttet 11 eu9eia"' = x-4' 11 01toia OtePXetat an6 'tO OTJJ.teio A .

ii) I:to O'T)J.tCio A tEJ.lVOVtat 01 eu9eie<; 0

tp"""- )o\m" tou cruan'Jpa'"'

PtPA.iou J.lC a = 100, P = 40.

{

X + 'JI = 20 , X- 'JI = 4. X+'JI=a (=100=p) _'I'= p = 40 =M� , 1t£p1ypacpet<11 crto 1tp6PA.TJJ.l<1 1, tou 1tprotou

(

)

EYKAEIAHl:. B' 89 T.l/7

------ B ' A\JKtiou KUl TU «nlaopanKU» npofll'll paTa TO\J Au)«pUVTO\J

-------

MOTJ nou napoucna�t'tat yta a = 20 Kat p = 4, aKoA.<m9ei TT)V 'ttXVtrit 'toov apt9J.tl'JnKrov:

on (0 J!tKpO'ttpoc; ) 'I' = c a (evac; apt9J.toc;), apa (0 J.lZYUA.'U'ttpoc; ) X = 4+ c a . X + 'I' = 20 ' £1tO�ooc; 20 = 2C a +4 ( Kai ano OJ.lOlWV OJ.lOla, a<j>alQW 4 ) 20 - 4 = 2C a + 4 - 4 2 C a = 1 6 . 'Apa 'I' = 8 Kat X = 12 ( Kai. TJ anobn�u; <j>aveqa ) iii) H <roJ.lJ!t'tptKft TT)<; 'I' = x-4, roe; npoc; TT)V 'I' = x , tivat l'J 'I' = x + 4 nou OttPXt'tat ano 'tO OTJJ.ttio

'Ecnro

{ OucnacrnKa otanpayJ.ta'teUt'tat Kat TT)V npo'taOTJ.

X -'1' = -4

..,11 = 96

iv) H elltt'l'l'J J.l£ tO"ritc; r(4,0), A(10.6) 1t0l) olipxt'tat U1t0 'tO OTJJ.ltlO A, txet t<pU1t'tO� TT)V ro9tia

'I' = X + 4. E<pa1t'tOV'tat O''tO OTJJ.ltlO E(3, 7), 61tou 'tEJ.lVOV'tUt l'J J.ltO'OKU9t'tO<; 'tO\) 'tJ.lTJJ.lU'tO<; rA Kat l1 ro9tia 'JI = X+ 4 . v) To op9oyc:Ovto Kat tO'OO"KtMc; 'tpiyoovo OEZ nou opi�t'tat ano 'ta OTJJ.ttia E, Z nou 'tEJ!Vtt l'J x+'JI=20 'touc; al;ovtc; Kat TT)V aPXft 0 'tOOV al;ovoov ' £xtt cJ.tPaoov 200. To op9oyc:Ovto Kat tO'OO'KtMc; 'tpiyoovo or� 1t0l) op�t'tat U1t0 'tU OTJJ.ttia r, � 01t0l) l1 'I' = X- 4 'tEJ!Vtt 'touc; al;ovec; Kat TT)V aPXft 0, exet tJ.tPaoov 8 (aepmcrJ.ta 208). 3) ( Al3aKT1Kt� an6\jlt:� )

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Ma9qpatta.oi Atayldvtopoi Ma9qpatta.is OA1lpnto6£S E1rtTpom} Ataycovt<Jp.rov Tll � E.M.E •

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Apyvpo ME'Tallw Apyvpo Mnall•o xaA.Kt.vo M.:Tall•o Et><p1JJI'I pvda Et><p1]JI'I pvda Et><p1]JI'I pvda

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MaDqpaTlKOi Awyrov1apoi - Ma01JpUTlK� OlupnuJ:o�

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An6 'tllV U1t06EOll 'tll � €nayroyi)� 'tO xropio am6 <5cv nepttxet a.'J..i.o Oll f.!ElO XPOOJ.la'to� KOKKtVOU .., f.l1tAE. Av, yta nap6.<5etyf.la, 'tO f.lOVa<5tK6 f.l1tAE Oll f.!Eio 'tou xropiou eivat 'to A, 'tO'te apKei va <ptpouf.!E J.lia w6eia nou <5taxrop�et 'to A an6 6A.a 'ta un6A.oma Oll f.lEia

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a.vricnOtxa.. 'Ecr'tro Em<Jfl� Ao 'tO JlEcrO 'tOU 'tO�OU /CB 'tQ'!> 19!KA.ou" n 1tO'Q 1tEPlEXE1 'tO A .

Av(i)...oya. op«;ouJ.LE Ka.1 'ta. <Jf1J.1Eta. B0 Ka.t C0 • A1to 'tfiV U1toes<Jfl_ .'to�,_1tpo�}dtJ.1a.'to� 'tO KEV'tpo,

'..

Ecr'tro Q ' 'tOU l<UKA.ou r ppicrKE'ta.t 1tllVO) <J'tOV �KUKAo � . L'tfl_ crl>VEXE10. ea. 0.1tObE�OUJ.lE JllO. / Portertnril 1tpO'tO.<Jfl . \ � . _ L At1 J.1Jla. Me m; V7roOeusZ<; rov 7rpofiA.Ijparoc;, eyovps Ao..B: = Ao(', . Em1rA.eov, \m -urfpda ..

.,-·

A, Ao , B1 , C, eivaz opoKVKA.uca. E7riurfc; m UYf)JBifx A Kaz

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fipipKo�az dTo it5zo 1Jple7ri7ret5o we;

1rpoc; 'l'1JV 1rkvpa B1C1 • AvriOTozxa uvp7rsprJ.�pam zufVovv Kaz rza rzc; a..u.Ec; t56o Kopv(pec; rov \ i / '' / \· rpzycbvov ABC . 1tou siva.1 A =;= Ao . "to'ts 'tO �piyrovo A1t60t1�11 A.t1ppaTo�. 0sropoUJ.1E 1tpflna. P1v 1ttpi1t'tro I

� ·

ABC siva.t 1crocrKEAE� cr'to A , 01tO'tE AB1 = AC1 Ta. u1toiJ,11ta. cr:t>J.11ttpcicr�a.'ta. sfVa.1 1tpo<pa.vft. ; . / 'Ecr'tro 'tropa. A * � . A�o 'tOV op1crJ.1? 'tO Ao txou � \'4,B =.f.4oC .1 E'lva.t s i<Jfl� tvrocr o 6n B1 C;'Ao · . i E J.!EVffi � ','ta. 'tp1yrova. BC, = CB, . E1t11tMov sxouJ.LE: LC,BAo = LABAo L;4- CAo AoBC, 1(0,� Ao OB, siva.t i a., ' 01tO)tE e� txduv K<l1 AoB, ! Ein1tJ..Eov 1tJo.fu1t 1 t<:at;11 cro'tflt6: r yrovtojv = o' tqv f �1t6ia. )11t , crl>J.11ttpa.f bu J.!E �n ''ta./ � Jl�a. A/,Ao , B1 ,/4;, siva.1

, f ·

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1 P Jl ou crs KA.o _jit Kopu<pe� �'tpa.1t'!J;u p_9u TElJ � J.Jtv buo a.ntva.vn 1tjfupe� l� A4o t<:a.1 B1 e; . o . ,_ E1ta.VEPXOJ.1EVQ1 / ;crtfl[ A.U<M 'toU /npo p).ft Jla.'to� na.pa.'tflpOUJ.lE on ·id J.lEia. 1 � , .B1 Ka.t C1 , I ppicrKOV'tO.l E<Jffi'tEpU�U cr . Kcl1t010 11J.1tKUdtKO 'tO�O 'tOU l<UKA.ou r ' 01t0t;E 'tO ,'tpiyrovo A,B,C, siva.1 a.JlPA.uyc.Ovto. Xrop� PA.ci P11 'tfl� 'YEV1KO'tfl'ta.� unoee'tOUJ.lE on 11 yrovia Kopu<pft� B, siva.t a.JlPA.sia..

'E'tcrt 'tO KEv'tpo Q Ka.t 'tO <JfiJ.!EtO B1 ppicrKOV'ta.t crs bta.<popsnKci

11J.1tsni1tsba.

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npo�

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�C1 • Av'ticr'tOtXa. crl>J.11ttpcicrJ.1a.'ta. tcrXt}ouv K<lt y1a. 'ta. <JfiJ.LEia. B Ka.t B" on6'ts 'ta. <JfiJ.LEia. Q Ka.t B ppicrKOV't0.1 <J'tO ilito 11J.11EmEbO ro� 1tp0� 'tfiV 1tAsupci A,C, . L11J.1E1c0VOUJ.1E on 11 J.LE<JOKcies'tO� 'tfl� � C, 'tEJ.1VE1 'tov l<UKA.o n crs buo <JfiJ.LEia. 1tou ppicrKov'ta.1 crs b1a.<popsnKci 11J.1tsmnsba. ro� 1tpo� 'tfiV �c, . An6 'tO 1tpcO'tO crl>J.11tEpa.crJ.1a. 'tOU AtlJ.lJ.l<l'tO�, 'tO. <JfiJ.lElO. Bo K<lt Q ea. siva.1 Kcl1t010. a.n6 't<l bUO <Jf1J.1Eta. 'tOJ.ltl�, 01tO'tE ea. crl>Jlm1t'tOUV, a.<pOU ppicrKOV't0.1 <J'tO tbto 11J.11Em1tEbO ro� 1tp0� 'tfiV A,C, . Tropa., 0.1t0 'tO 1tpcO'tO J.!Epo� 'tOU AtlJ.lJ.lO.'tO� 1tpOKU1t'tE1 on 01 suesis� QAo Ka.t QCO siva.t 01 J.LEcrOKcies'tt� 'trov 'tJ.111Jlcl'trov B1C1 t<:a.1 A1B1 , a.v'ticr't01X<l. E1toJ,Jtvro�

LC1B0� = LC1B0B1 + LB1B0� = 2 (LAoB0B1 + LB1B0C0 ) = 2 · LAoB0C0 = 1 80° - LABC, (1) a.cpou 'ta. <JfiJ.LEia. Ao Ka.t C0 siva.t 'ta. J.!Ecra. 'trov 't6�rov CB t<:a.1 BA , a.nicr'tOtXa.. EYKAEIAm: B' 89 '1'.1/12

------

MaOfiJUlnKoi Auryrovurp.oi - MaOflp.anK� Ol..1>p.mao� ------

E1tutAtov, a1t6 'tO bcU'ttpo J.!Spoc; 'tO'l> J.:iJJ..tJ..ta'tOc;, sxouJ..tt :

LC1B0A1 = LC1B� = LABC .

(2)

A1t6 ne; (1) Kat (2) A.aJ..t�civouJ..tt 6n LABC = 9 0 ° . 2°.; Tpo1toc; [At>Gll llavayuo'"l Al}JUIK1J] 1\flJ..t f.l a. 'Ecr'tro tpiyrovo ABr Kat OT)J..ttia /1, E 1tcivro crnc; 1tA.tupsc; 'tou AB, Ar , aV'ticr'tmxa, 'tS'tota fficr'tt B/1 =TE . T6'tt 11 J..ttcroKaet'tT) 'tT)c; Br Kat 11 J..ttcroKaet'tT) 'tT)c; !1E 'tSJ.!VOV'tat 1tcivro cr'tov 1ttptytypUJ.!J.!SvO ri>do 'tou 'tptyffivou ABr . A1t63ct�ll· 'Ecr'tro 6n 11 J..ttcroKaet'tT) 'tT)c; 1tA.tupcic; Br 'tSJ.!Vtt 'tOV 1ttptycypaJ..tJ.!Svo ri>KA.o cr'to OT)J..ttio M . T6'tt sxouJ..tt Mr = MB, B/1 = rE Kat LMr A = LMBA roe; tyytypaJ..tJ.!Svtc; cr'to i3to 't6�o. E1toJ.!Svroc; 'tU 'tpiyrova MrE Kat MB/1 tivat icra, 01tO'tt ME = M/1 . A.pa 'tO 'tpiyrovo MilE tivat tcrocrKtAtc;, o1t6'tt 11 J..ttcroKaet'tT) 'tO'l> /lE 1ttpvcitt a1t6 'tO <n]J..ttio M . 0

E1tavtpx6J..ttvOt cr'to 1tp6�AllJ.!U J..tac; 1tapa'tT)pOUJ..tt 6n 'tO 'tpiyrovo A1B1C1 tivat aJ..t�A.uyffivto, a<pou

bta<popt'ttKcl 'tO 1ttptKtv'tpO 'tO'U 0 btv ea avflKt cr'tOV 1ttptytypUJ.!JlSvO KUKAo Q 'tO'U 'tptyffiVO'U ABC . Y1toes'tO'UJ.!t O'tt 'tO 'tpiyrovo A)Bpl tivat UJ..t�Auyffivto cr'tO Al . T6'tt 'tO 0 ea avflKtt (j'tQ avriet'tO T)JlltiD1ttbO 'tT)c; C) B) U1t0 amo 1tO'U avflKtt 'tO OT)J.!ttO AI .

Em1tMov, a1t6 'tO AtlJ.!J.!U 1tpori>1t'ttt 6n 11 J..ttcroKaet'tT) 'tO'l> C1B1 'tSJ.!Vtt 'tO 'tO�o BA C cr'to J.!Scro 'tO'l> 1tou �picrKt'tat crt 3ta<poptnK6 T)J..ttt1tt1ttbo J..tt 'tO A1 roe; 1tpoc; 'tT) C1B1 • Em1tMov yvropfSouJ..tt O'tt 'tO 0 (1ttptKtv'tpO 'tO'U 'tptyffivou AI B) cl avflKtt cr'tT) J..ttcrOKaet'tT) 'tO'U cl B) ' 01tO'tt . 'tO 0 tivat

'tO J.!Scro 'tO'l> 't6�ou BA C . Tffipa 'ta 'tpiyrova OB1C Kat OC1B tivat i<;fa, a<pou sxouv LOCB1 = LOBC1 roe; cyytypUJ.!J.!Evtc; -

cr'to i3to 't6�o, OC = OB, a<pou 'tO 0 avflKtt cr'tT) J..ttcroKaet'tT) 'tT)c; BC Kat CB1 = BC1 • A.pa SXO'l>J..tt 'tT)V tcrO'tT)'t«l yrovtffiv LC10B = LB10C � LC10B1 = LBOC = LA . E1tutAtov T) J..ttcrOKaet'tT) 'tO'U AIBI 1ttpvcitt a1t6 'tO 0 Kat 'tSJ.!Vtt 'tOV ri>KA.o

cr'tO OT)J..tttO K '

J..tscro 'tO'l> 'tO�ou ACB . OJ..toiroc; 11 J..ttcroKaet'tT) 'tO'l> C1 A1 1ttpvcitt a1t6 'tO 0 Kat 'tSJ.!Vtt 'tOV ri>KA.o -

cr'to OT)J.I£tO L , J.!Scro 'tO'l> 'to�ou ABC . E1toJ.!Svroc; sxouJ..tt LC LA LLOK = l 8 00 - AZ - :4K = l 8 00 - LA

I II I

�

= 9 00 -

�

DJ..troc;, LA10L = LC10L, a<pou 11 LO tivat J..ttcroKaet'tT) 'tT)c; C1A1 Kat OJ..toiroc; LA10K = LKOB1 , 01tO'tt LLOK = _!_ LC10B1 � 90 ° -

•

= _!_ LBOC =

� LA = 90°.

EYKAEIAm B' 89 -r.l/13

(I)

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I < ( I ) f 3YiaoA�grin"t

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{vJdvl:raog)

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(I)

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l(l')

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MaOqpo:nKoi Auryrovurpoi - MaOqpanK� 0A1>pnu1ot� -----n

n E

I ( ) ;::: nl ( 1) ;::: yta K<iOs z>O (2) m (Q> sxouJ.t.s I ( m ) I ( ) :::: I ( m) :::: m > 0 , on6�s Eniall <; , an6 'tT)V (i) yta K<iOs q 0 I ( q ) > 0, yta K<iOs q (Q>0 . (3) lotattspro<; EXOUJ.lS

(ii) �c:Opa npotcin�st 6n TJ cruv<ip'tT)<Tll I sivat 'YVTJcriro<; m'>�ouaa, a<pm) yta x , y E Q>0 , taxl> st 6n x - y > 0 Kat I ( x ) = I ( x - y + y ) :::: I ( x - y ) + f ( y ) > f ( y ) , 'A6yro 'tT)<; (3) .

An6 'tT)

J.1S x >

I (q) q ' �6�s sUKOAa <ruJ.t.1tSpaivoUJ.lS �0 ioto yta K<iOs esnK6 PTJ�O X p L'tT) cruvsxsta 1tapa'tT)pO'UJ.lS 6n s<p6crov anoos�Ol>JlS 6n

yta K<iOs esnK6 aKspato

a<pm) tcrxl}st:

f ( x ) q = I ( x ) I ( q ) :::: f ( xq ) = I ( p ) = p = xq => f ( x ) :::: x , ( 'A6yro 'tT) <; ( 1 )) : qf ( x ) ::;; I ( qx ) = I ( p ) = p = qx => f ( x ) ::;; x , 01tO�S an6 n<; (4) Kat ( 5) 1tpOtcU1t�St TJ tcr6'tT)�a I ( X ) = X ' yta K<iOs esnK6 aKspato =

(4) (5) X.

I (m) m, yta K<iOs OutK6 aKspato m . A<; unoesaouJ.lS 6n un<ipxst esnK6<; aKspato<; m �S�oto<; c:Ocr�s I ( m) > m. T6�s ea SXOUJ.l.S > 0 Kat unapxst Ka�<iMTJAa J.lSY<iAo<; esnK6<; aKspato<; N �S�oto<; c:Ocr�s f ( m) m + N& > 1 . T6�s EXOl>J.t.S 6n I ( Nm) :::: Nl ( m) N(m + &) > Nm + 1 . H �s'Asu�aia crxsall a'A1JOs6st yta K<iOs aKspato JlS'YnA1hspo LQ'\) Nm a<pm) I ( Nm + p ) :::: NI ( m) + p > Nm + p + 1 . E>sropc:O�a<; OJ.t.OO<; �c:Opa �0 aUVOAo �(OV aptOJ.t.roV as's z >0 ' 1tapa'tT)pOUJ.lS 6n un<ipxst J.tia ( 6) OUVaJ.t.TJ �0'\) a �s�ota c:Ocr�s as > Nm. lotat�spro<; SXOl>J.lf: 6n: I (as ) ;::: I ([as ]) ;::: [as J + 1, 6nou J.1S [as J <ruJ.t.�o'AfSoUJ.t.S m aKspato J.t.Spo<; �ou as MnopOUJ.lS OJ.t.OO<;, an6 'tT)V un60sall (iii), va anoosi�Ol>J.lf: 6n I (as ) ::;; f ( a r as ' (7) 01tO�S a1t6 n<; (6) Kat (7) 1tpOtcU1t�St on as ;::: [as ] + 1, 1tOU StVat a�01t0. EnoJ!Svro<; apKsi va anoosi�OUJ.lS 6n =

E' E

llpoP'AlJJla 6. 'Emro n :::: 3 tva<; aKtpato<; apt9po<;. 9t:ropovpt: tva K't)K'Ao pt: n + 1 GtJJ1Eia 1tUVro CJE amov 1t01> xrop�Ol>V TOV KVKAo CJE n + 1 iCJa To;a. 9t:ropovpt: OAE<; Tl<; GlJJlUVCJt:t<; auTrov Trov GtJJlEirov JlE T01><; apt9pov<; 0,1, , n ETCJt romt: Ka9t: apt9J16<; va XPlJGtJlonott:iTat Jlia JlOVo (J)Opa. Avo TETott:<; GlJJlUVCJt:t<; 9t:ropovVTat oTt Eivat iott:<;, av lJ pia J11topEi va 1tpOKV'JIEt ano TlJV UM1) pt: pia 1tt:ptmpo(J)1] T01> KVKAol>. Mia CJ'I]JlaVGlJ ovopa�t:Tat mpaia, av yta onotol>CJO'I]noTt: TECJCJt:pt<; apt9J10V<; a CJ1>VOEt:t Ta GtJpt:ia JlE TlJ CJ'I]p«VGlJ a Kat d Ot:V TEpVt:t TlJ xopo'l] 1t01> CJ1>V0Et:t Ta GtJpEia pt: TlJ CJ'I]paVGlJ b Kat C. 'Emro M o apt9J10<; Trov ropairov GlJpUVCJt:rov Kat ECJTro N o apt9J16<; Trov otaTt:Tayptvrov �t:l>yrov ( x, ' 01t01> x, 9t:TtKOi aKtpawt, ETCJt romt: X + ::;; n Kat MKA ( x, = 1 . N a • • •

anooEi;t:Tt: OTt: M = N + 1 . AVGlJ �sooJ.lSvTJ<; J.tta<; KUcltKft<; <rflJ.t.aV<Ttl<; �rov aptOJ.t.rov mu cruv6'Aou

[Pmuia 1

{ 0, 1,

..., n

} := [ 0, n ] ,

opi�ouJ.lS

J.tia k - xopoi) �ou Kl>K'Aou SKStVTJ 'tT)<; onoia<; �a ouo <iKpa (ntOavc:O<; �aun�6J.1SVa) sxouv aOpotaJ.ta <Tll J.lUV()f:(OV iao JlS k . AEJ.t.S 6n �pet<; xopos<; �0'\) tciclou sivat suOuypaJ.t.J.ll()J.l.SvS<;, av J.tia an6 au�s<; otaxropil;st n<; <iUs<; ouo as ota<popsnKa AEJ.lS 6n 3 xopM<; �ou tciclou sivat suOuypaJ.t.J.t.tcrJ.t.SvS<;, av onotsaoi)no�s �pst<; an6 ams<; sivat suOuypaJ.t.J.t.tcrJ.l.Svs<;.

m ;:::

EYKAEIAHI: B' 89 'T.l/15

------ Ma9qpanKoi At.ayrovuJpoi - Ma9qpanK� Ol..1>pma�� rta 1tapa0£t'YJ.la <J'tO axf!J.la 1 1tOU alCOAoU6si ot xopotc; A, B Kat c sivat su6uypaJ.lJ.ltcrf.l£vsc;, svro

-------

ot xopotc; B, c Kat D 0£\1 sivat. Al}JlJla. 'Ee Jlla wpaia af!JtaWJ1'f oz k - xopt5b; eivaz evOvypaJlJlU1JleVe' yza KafJe aKtpazo k . A1t00Et�11 . ea XP11crtJ.l01totftcrOUJ.lS £1tayo:Yyi]. To S11'tO'UJ.lSVO t<JXUst 1tpO<pavroc; yta n :::;; 3 . 'Ecr'tCO 'tcOpa n � 4 . A<; 6sropftcrOUJ.lS J.lla ropaia crtlJ.lW<Jll <J'tT}V 01toia 'tptt<; k - xopOtc; A, B Kat c osv sivat su6vypaJ.lJ.ltcrf.l£vsc;. Av 0 apt6J.lO<; n 0£\1 sivat Ka1tot0 a1to 'ta aKpa 't(I)V xoporov A, B Kat c, 'tO't£ J.lS Otaypacpft 'tOU

S A.aJ.tPavouJ.lS

a1to 'to cr6voA.o

S-

J.tia ropaia crftJ.laV<Jll

{ n}

'tOU

[ 0, n - 1] ,

o1tou ot xopotc; A, B Kat C sivat su6uypaJ.lJ.ltcrf.l£vsc; a1to 'tT}V u1to6s<J11 'tT}<; t1tayro'Ytlc;. OJ.toiroc;, av 0 apt6J.lO<; 0 osv sivat Ka1tot0 a1to 'ta aKpa 't(I)V xoporov A, B Kat C, 'tO't£ J.lS Otaypacpft 'tOU 0 a1to m cr6voA.o S Kat J.lStrovoV'tac; oA.ouc; 'touc; apt6J.1o6c; Ka'ta tva A.aflP avouJ.lS Jlia ropaia crftJ.laV<Jll s - { 0} 'tOU [ 0, n - 1] ' 01tOU Ot xopotc; A, B Kat c sivat su6uypaJ.lJ.ltcrf.l£vsc;. E1tof.l£vroc; Kat 'tO 0

Kat 'tO n sivat Ka1tota a1to 'ta aKpa 'tCOV xoporov A, B Kat c . Av x, y sivat 'ta av'tt<J'totxa 'tOU<; OsU'ttpa aKpa, 'tO't£ 6a txoujl£ n � 0 + X = k = n + y � n . A.pa 0 Kat n sivat 'ta ouo aKpa J.ltac; xopoftc;, tcr'tro C .

I:XJ1J.la 2 I:XJ1J.la 1 'Ecr'tCO D 11 xopOft 1tOU crxrtJ.lanl;t'tat a1to 'tOU<; apt6flOU<; u Kat V 1tOU sivat yst'tOVtKOi 't(I)V 0 Kat n Kat ppicrKOV'tat 1tpoc; <J'tO ioto ,.tpoc; 'tT}<; xopoftc; c ' 01tCO<; Ot xopotc; A Kat B <J'tO axf!Jla 2. et'tOUJ.lS t = u + V . AV tl'taV t = n ' 'tO't£ ot n xopotc; A, B' Kat D 0£\1 6a sivat su6uypaJ.lJ.ltcrf.l£vsc; cr'tT}v ropaia crftJ.laV<Jll

S-

{ 0, n} , �'t01tO. Av sivat

n , 'tO't£ 11

t - xopoft J.1S

aKpa 'tO 0 Kat 'tO t 0£\1 J.mopsi va 'tEJ.lVSt 't1} xopoft D ' 01tO't£ 11 xopoft c Otaxropil;st 'tO t Kat 't1} xopoft D . H xopoft E JlS aKpa t Kat n - t 0£\1 'tEJ.lVSt 't1} xopoft c ' 01tO't£ Ot t Kat n - t sivat 1tpoc; 'tO ioto J.lEpoc; 'tT}<; xopoftc; c . To't£ OJ.lCO<; ot xopotc; A, B Kat E 0£\1 sivat su6uypaJ.lJ.ltcrf.l£vsc; <J'tO cr6voA.o t

{ 0, n} '

a't01tO. TtA.oc;, 11 1ttpi1t'tCO<Jll

,.tcrro 'tOU J.lS'tacrxrtJ.lancrJlOU

Kat cr'tEAVSt ne;

t - xopotc;

crn<;

�

sivat tcroouvaJ.l11 J.lS 'tT}V 1ttpi1t'tCO<Jll

n - x, 0 :::;; x :::;; n, o o1toioc;

( 2n - t ) - xopotc;.

Ota'tT}pti

ne;

ropaisc; <Jll J.laVcrtt<; D

I:'tT} cruvtxsta yta 'tT}V a1toOst�11 'tou 1tpopA.ftJ.la'to<; 6a 1tpoxropftcrouJ.lS J.1S t1tayro'Ytl. H 1t£pt1t'tCO<Jll n = 2 sivat 1tpocpavt1c;. A<; U1to6tcrOUJ.lS on n � 3 . esropOUJ.lS Jlla ropaia crftJ.laV<Jll s 'tOU [ 0, n] Kat otaypacpouJ.lS 'tO n , o1to'ts A.aJ.tPavouJ.lS J.tia ropaia crftJ.laV<Jll T 'tou [ 0, n - 1] . Ot

n - xopoec; 'tT}<;

T sivat su6uypaJ.lJ.ltcrf.l£vsc; Kat 1ttpttxouv Ka6s <Jll J.lSlO SK'tO<; a1t6 'tO 0. Atjl£ on 11 crftJ.laV<Jll T sivat W1COV 1' av 'tO 0 ppicrKt'tat J.lS't�U ouo a1to au'tt<; ne; n - xopOtc;, svro au'tft sivat W1COV 2, crs Ka6s OtacpopsnKft 1ttpt1t'tCO<Jll . ea a1tOO£i�OUJ.lS on Ka6s crftJlaV<Jll W1COV 1 'tOU [ 0, n 1tpotpxs'tat a1tO J.lta J.lOVO ropaia crftJ.laV<Jll 'tOU [ 0, n ' svro Ka6s crftJ.laV<Jll W1COV 2 'tOU

- 1]

]

EYKAEIAHI:. B' 89 1:.1/16

------

Ma91)p.anKoi Alayrovwp.oi - Ma8qp.anK� OA.vp.m46�

-------

[ 0, n - 1] 1tpoepxeiat a1to aKpt�ro� ouo ropaie� CTI'JJ.uxvcret� -rou [ 0, n] . AV 11 T eivat rV7rOV 1 ,B<J't(l) O'tt 'tO 0 �picrKetat J..teta9} 't(I)V xoporov A Kat B . E1tet0ft 11 xopoi) a1to to 0 crto n 1tpe1tet va eivat eu&uypaJ.l�ucrJ.livrl J..te n� xopoe� A, B tou S , to n 1tpe1tet va .

�picrKetat crto Cxuo to�o J..tet�u trov A, B . E1toJ.l8vro� to <JUvoA.o S J.11tOpei va 1tpo8A.6et J.lOVOcri)J.lavta a1to to <JUvoA.o T . �ta<popsnKa, av 11 T eivat r61rov 1 Kat etcrayouJ..te to n o1tro� 1tapa1tc:ivro, tote 11 crftJ.laVCTI'J 1tou 1tpocintet eivat ropaia. IlpayJ.lan, txouJ..te : • Av 0 < k < n, 'tOte Ot k - xopoe� 'tOU s eivat emCTI'J� k - xopoe� 'tOU T ' 01t0Te e{vat su&uypaJ.1J.1tcr J.l8ve�. • Av n < k < 2 n ' OeooJ.18vou O'tt ot n - xopot� 'tOU s eivat 1tapa.A.A.11A.e� a1tO 'tllV Kata<JKeuft 'tOU�, U1tclPXet eva� a�ova� <JUJ.lJ..tetp{a� £ tB'tOtO� rocrte Kcl6e X . eivat cruJ.lJ..te'tptKO 1tp0� to n - X (I)� 1tp0� 'tOV £ ' yta Ka6s X Av U1tapxouv ouo k - xopoe� 1tOU 'tBJ.lVOV'tat, 'tOtS ot <JUJ.lJ..te'tptKB� 'tOU� ro� 1tp0� 'tOV a�ova £ 6a etVat OUO ( 2n - k) - xopOe� 1tOU 'tBJ.lVOVtat J..te •

0 < 2n -k < n , ato1to. Av 11 T eivat rV1rOV 2 ' }6�e U1tapxouv ouo ouvate� 68cre� yta 'tOV n <J'tllV s ' J.l{a <Je Ka6e 1tA.eupa tou 0. 01tro� 1tapa1tavro J.11topouJ..te va oouJ..te on Kat ot ouo aute� 68cre� oo11youv cre ropaie� CTrJJ.!clVcre� tou [ 0, n] . E1toJ.l8vro�, av Mn eivat o apt6J.1o� trov ropairov CTI'JJ.lclVcrerov tou [ 0, n]

Kat Ln eivat o apt6J.1o� trov ropairov CTI'JJ.lclVcrerov tou

[ 0, n - I] r61rov 2, tote txouJ..te :

Mn = Mn-l - Ln-l + 2Ln-l = Mn- 1 + Ln-l

(

)

A1toJ.l8vet va a1tooe�OUJ..te on Ln_1 eivat o apt6J.16� trov �euyrov Kat gcd ( x, y) = 1 . E1tet0i) eivat

( x, y)

6enKrov aKepairov J..te

n � 3 , o apt6J.1o� auto� tcrootat J..te tp ( n) , 011A.aoi) J..te tov apt6J.1o trov 6ettKrov aKepatrov x J..te 1 � x � n Kat gcd ( x, n) = 1 . fta 'tllV a1toOe�11 autou 6eropouJ..te J.lia r61rov 2 ropaia crftJ.laVCTI'J tou [ 0, n - I] . l:taJ.11tapouJ..te t� 6tcre� trov apt6J.16>v 0, ... , n - I ( mod n) crUJ.l<prova J..te 'tllV KiVllCTI'J tO)V OetKtrov tou ropoA.oyiou, etm rocrte o apt6J.1o� 0 va �picrKetat <J'tll 68CTI'J 0. 'Ecrtro f ( i) o aptHJ.lo� 1tou �picrKetat <Jtll HtCTrJ i . l:11J..tetrovouJ..te on 11 f eivat J.lia J..tetaHsCTrJ tou [ 0, n - I] . 'Ecrtro 11 HtCTrJ 1tou eivat tetota rocrte f (a) n - I . E1tetOi) Ot n - XOPOB� eivat eu6uypaJ.1J.1tcrJ.18ve� J.le tO 0 Kat Ka6e CTI'JJ..telO avi)Ket <Je Kcl1tota n - XOpOi), e1tetat on aute� ot xopoe� eivat 1tapa.A.A.11A.e� Kat f ( i) + f ( -i) = n, yta KaHe i. OJ.1oiro� enetOi) ot ( n - L}-/xopoe� eivat eu&uypaJ.1J.1taJ.l8ve� Kat KaHe CTI'JJ..teio avi)Ket cre Ka1tota ( n - I) - xopoi), ot xopoe� aute� eivat 1tapaA.A.11A.e� Kat t<JXI)et: f ( i) + f (a - i) = n - I, yta KaHe i . E1toJ.l8vro�, txouJ..te f (a - i) = f ( -i) - 1, yta KaHe i Kat e1tetOit f ( 0) = 0 , AaJ.l�avouJ..te f ( -ak) = k, yta KaHe k . ( 1) 9uJ.1�0UJ..te ott 11 CJXBCTI'J (1) eivat o J.1ia t<JO'tllta modulo n , ooote acpou 11 f eivat J.lia J..tetaHeCTrJ, Ha 1tp81tet va txouJ.lS gcd ( a, n) = 1 . E1toJ.1Bvro�, Ln-l � tp ( n) . x+

y=n

•

fta 'tllV a1tOOe�11 'tll � t<JOtll 't�, a1toJ.l8vet va a1tOOe�OUJ..te O'tt 11 aftJ.laVCTI'J 1tOU oivetat a1tO 't11 CJXBCTI'J ( 1) eivat ropaia. IlpayJ.latt, av Heropi)aouJ..te teaaep� aptHJ.lou� w, x , y, z 1tavro atov clKA.o etat roate w + y = x + z , tote ot Heae� tou� atov clKA.o tKavo1totouv 'tllV tCJOtll ta

(

-ax

-az

) + (-ay) = ( ) + ( ) y Kat 11 XOpOi) a1t0 tO

tO 01t0l0 CTI'JJ.la{Vet Ott 11 XOPOTt a1t0 'tO W OtO X OtO Apa 11 (1) eivat ropaia crftJ.laVCTI'J Kat a1to 'tllV Kata<JKeuft 'tll � eivat r61rov 2 . EYKAEIAHl: B' 89 "r.l/17

eiVat 1tapc:iA.A.11A.£�.

----....--... Ma&qpanKo( A1.aymvurpot - Ma&qpanK� 0/..'U pn1.a3� -----

3 011

BaA.KavtKll Ma9rtJ.lUT1Kll OA.vp1tul�a Kt>npoc;, Aypoc;, 28 lot>viot> troc; 3 lot>J..iot>2013

H 30'� BaA.KavtKil MaHru..tanKil OA.uJ..lmaoa f:ytve crtov Ayp6 tll <; Ku1tpou, a1t6 2 8 Iouviou too<; 3 IouA.iou 20 1 3 . 'EA.a�av J..ttp o<; <.TUVOAtKcl 17 xropt<;, l;uJ..l<pOOVa J..lE tOV KaVOVt<.TJ..lO E1ttcnlJ..lOO <; <.TUJ..lJ.E.l teixav ot xropt<;: AA.IJavia, Boavia-Ep�qoiJiVfl, Bo'UA.yapia, EUclaa, K'6npo�, Ma'Upo1Jo'6vto, MoA.aaiJia, nrA Tl)<; MaKtaovia�, Po'Upavia, I:.tpiJia, tvro avemcrJ1J..lro<; <.TUJ..lJ.E.l teixav Ot OJ..lclOE<; t(l)V xroprov: A�tppnai'T�av, Hvmptvo Ba<rwto, haA.ia, Ka�aKG'TUV,

Ta'T�tKtG'Tav, To'UpKptvtG'Tav Kat To'UpKia B. H EM11VtKil OJ..lclOa KattKtll <.TE 1 XP'UGO JlE'TUMtO, pt'TO.Uto, ro<; t�f1<;: AroA.ac; Davayt<l>TI} c; A11J10.Ktt <; DavaytTI)<; T<rivac; Kmv<r'Tav'Tivoc; N'To'6Vf1<; Dt'Tpoc; l:.Ktaaono'UA.o<; A011vayopac; Aa�ap11<; E't>ayytl.oc;

ap"('UpO Jlt'TUMtO Kat 3 xaA.Ktva

EK7t. TptKO.A.mv A0HNA I:.xoA.l} MmpaiTI) 1° rEA TptKaA.mv 1° rEA Kopmnio'U Poaimv Datada A'6Ktto E'UayytA.tKl}� I:.xoA.l}c;

Xp'U<rO pt'TO.Uto Ap"('Upo pt'TO.Uto xaA.Ktvo pt'TaUto xaA.Ktvo pt'TaUto xaA.Ktvo pt'TaUto I:,'U J1Jlt'TOXll

l;uvoooi trov J..la9rttrov f1tav o AtKtopa<; [u1t6 otoptcrJ..l6] tou E. M. TioA.utex;veiou K. A11J11\'Tptoc; KOV'TOKIDG'Ta� Kat 0 J..la9rtJ..la'ttKO<; K. rtropytoc; A7tOG'TOA07tO'UAo<;. Ot J..la811tt<; otayrovil;ovtat crt ttcrcrepa 1tpo�A.t1J..lata 1tou Ka8tva a1t6 auta �a8J..loA.oyeitat J..lE 10 J..lovaoe<;. l;uJ..l<prova J..lE tllV atu1t11 �aSJ..loA.oyia 11 EUaoa J..lE 89 J..lOvaoe<; KattA.a�e tllV 411 Stcrrt J.!Eta�u tOOV E1ttcnlJ..lOO <; <.TUJ..lJ.t.lf: EXOU<.TcOV xroprov, tvro f1tav <.TtllV 711 8tcrrt J.!Eta� tOOV 17 OJ..laOOOV 1tou cruvoA.tKa tA.a�av J..ttpo<;. Tt<; tptt<; 1tprott<; 8tcret<; KattA.a�av ot OJ..laOt<; tll <; PouJ..tavia<; (195), BouA.yapia<; (149), Kat l;ep�ia<; ( 1 32 ) .

Ta npopJ..t] paTa

DpoiJI...TtJUl 1. I:.t tva 'Tpiymvo ABC 0 napqqpapptvoc; K'6KA.oc;

antvavn ano TI)V KOp'U<pl}

A t<pamt'Tat Tl)<; AB <r'TO P Kat Tl)<; A C G'TO Q , Kat o napqqpapptvo� K'6� wb antvavn : ano 'T'IV KOp'U<pl} B t<pamt'Tat 'T'I<; BA <r'TO M KaJ 'T'I� BC <r'TO N 'E<r'Tm K 11 op91} npoiJol.l} 'TOO C mi 'T'I<; MN Kat t<r'Tm L 11 op91} npoiJoA.l} 'TOO C tni 'T'I<; PQ . Na anoattxQd 6n 'TO •

'Tt'TpanA.tu po MKLP dvat E'Y"fPU'I/lJlO <rt K'6KA.o. A6<r11. (1 oc; Tponoc;)

E1tetof1

L.KMP = 90° -

B 2

(l;xflJ..la

1 ),

apKti va a1toOt�OUJ..lE 6n L.KLP = 90° +

B 2

B . 'Ecrtro I ' to tKKtvtpo tou tptyrovou ABC Kat tcrtro D to crrt J..lEio E1ta<pf1<; tou 2 f:'Y'YtypaJ..lJ..ttvou cidou tou tptyrovou ABC J..lE t11V 1tA.tupa AB ; E1ttt0tl CK 1 1 IB Kat CL 1 1 lA , B L.KCL = L.AIB Kat L.KCN = L.IBC = t1tttat 6n: 2 b+c-a E1tt1tA.tov a1t6 CN = AD = -r - a = , (BC = a, CA = b, AB = c), 61tou 2-r = a + b + c eivat 11 2 B B B A 1ttptJ.!Etpo<; tou tptyrovou ABC , A.aJ..l�clVOUJ..lE : CK = CN cos - = AD cos - = AI cos - cos - . 2 2 2 . 2 L.KLC =

�

EYKAEIAHI:. B' 89 T.l/18

------

Mu&qpunKo( Aurymv\Gpot - Mu&qpunK� 0).'Upm.6.6�

Av(i)...oya 7tp01c6xtst 6tt CL = BI cos

A 2

cos

Kat AIB dvat 6J.Lota, 07t6ts 9a f:xouv Kat

B 2

o7t6ts:

AI .

-------

E1to�co; , ta tpiycova KCL

LKLC LAB! B , 61tco; 9tA.a.J.LS. =

-rpo1to<;. 'Ecrrco T = MN n PQ (IxfuJ.a 2). ApKsi va a7toSsi�OUJ.LS 6n TC 1. AB , aq>m) t6ts 2°�

Llv!PL = LTCL = LTKL = 1 80" - LMKL , o7t6ts to tstp6.7tA.supo MKLP 9a sivat syy pawtJ.Lo. 'Ecrrco H 11 7tpopoJ.:t; tou c 7tlt.vco CTtllV AB. Kat PR J.Lia StaJ.Lstpo; tou clrl.ou tv0 H OJ!Oto9saia h dvtpou •

C 11 07tO{a a7tStKOV{�St tOV clrl.o

aJ0

CTtOV tvb , a7tStKOV�St SmO'll ; tO O'll J.LSiO crro M. llpayJ.Lan, 11 h crrtA.vst tl1 StaJ!Stpo PR tou OJ0 as 1tapa.M11A.11 su9da Kat R

rub .

Stlt.J.Lstpo tou

Aq>m) 11 PR dvat

Ka9stl1 CTtllV AB , 11 stK6va tou PR 9a dvat 11 Ka9stl1 7tpo; tllV AB a7t6 to KMpO 02 tOU rua , 07tOtS 11 SUCOVa tOU R

sivat sits 11 7tpoPoA.ft M tou Ktvtpou tou OJ0 CTtllV AB , sits to

avttStaJ!StptKO tOU CTtOV

sivat to M =

Kat M tcov O'll J.Lsicov R, C Kat M 1tavco CTtllV AB tKaV07tOtOUV tl1 <JXEO'll : PH

Pa . 'Ecrrco Pb

7tavco CTtllV

AB,

11 7tpopoA.ft tou T

----

Tr. ----tl'

- -- -

to Ktvtpo tou

7tapsysypaJ.LJ.Ltvou clrl.ou a7ttvavtt a7t6 tllV Kopuq>ft C Kat tcrrco S to O'll J.LSio s7taq>ft; tou we J.LS tllV 7tA.supa AB. E7tstSft ta tpiycova PTM Kat

AIcB

f:xouv 1tapa.M11A.s; 7tA.supt;, dvat OJ!Ota, evcb Suo avticrrOtxa uw11 tou; sivat ta TG Kat IcS . E7toJ.Ltvco; txouJ.LS

d(C, wa ) Pa GM d( C, wb ) Pb d ( C, ) Kat d ( C, wb ) sivat PG

AS SB

-- = - =

tv0

•

<J)umKa

!l; auvt7tsta ta O'll J.LSia R, C Kat M va sivat auvsu9staKa Kat S7tt7tA.tov

C Pa . E7toJ.Ltvco;, ot 7tpopoA.t; P, H CM Pb R

rua

----

07t0U

= -,

llTtKll

tCOV Sq>a1t'tOJ.LSVCOV tJ.L11J.LatCOV a7t6 tO

EYKAEIAHl: B' 89 -r.l/19

---

.. - -0

•,

...

.. ...

...

.. ...

-- --. o_; --

----

------

1tp0� -rou� JC61CA.ou� 3 o.;

lV0

OJ.10ico�

Kat aJb , avricnOtxa. Apa txouJJ£ -reA.tKa

PH HM

-rpo1to�. 'Ea-reo D -ro mtJJ£io 'tOJ.u't� -reov MN Kat

LCDQ = x, LCDN = y .

LBMN 90· =

-------

Mue,.panKo{ Aury(I)VUJpo{ - Mue,.pun� 0A.t)pnuJ6� Pa

Kat 6-rt

H=G.

PQ (txflJ.La 3) Kat to-reo A Eut5ft a1t6 o taocnceA.t� -rpiyeovo APQ eivat LAPQ 90• Kat B

t1te-rat 6n

x+y

A + B . t'tT) auvtxeta JJ£ -ro auvft9t) OUJ.1�oA.taJ.16 .

2 2 Kat -ro v6J.10 teov TJJ.1tt6veov o-ra tpiyeova CND Kat CQD txouJJ£: r-a CD r-b CD Kat = --

ov v -

01tOTE

AaJ.1�clvOUJ..I£

ovv

ov v

A 2 -

1'/f.JX

B - 1'/f.JY 2

1'/f.JY

1'/f.JX

ov v -

= --

urp - 1'/f.JX . B2 b 1'/f.JY urp '

(XPTJatJ.101tOtftaaJ..I£

. _

r-a

AT BT

r-b

-- = - =

A urp 2

B urp -

't�

axtae�

, T etvat -ro

, 61tou

2 mtJJ£io E1tacpft� -rou eyyeypaJ.1J.Ltvou JC61CA.ou J..1£ 'tT)V 1tA.Eupa AB ), 01t6'tE

17f.J 1'/f.JX --t . A1t6 'tT)V 1'/f.JY 1'/f.J -

1tpOJC61t'tEt 6n taO'tT)'ta +

x y Y ' ' ' ' ' ' '

au'tft

A+B

2 'tT)

Kat

e1te-rat 6n

B (av 9e<JOUJJ£ 2 'tT)

f.JY

01 ,' '

1'/f.J B 2

X+y

y ) 'I,U ( t -%\ q,u(t1'/

txouJ..I£

axtmt

= -

Kat 'tOTE

axtmt

Elto�,

txouJJ£ DC .L AB Kat, av cl eivat 'tO mtJJ£io TOJ.1Tt� -reov CD Kat AB , txouJ..I£ DL · DP = DC · DC1 = DK · DM , 01t6-re -ra mtJJ£ia M, P, K Kat L eivat OJ.10KUKAtKa. txflJ.La 3

Ilpofll..f1 J.1a 2. Nu 1Jpt:9oov 6A.cn 01. 9enKoi aKtp•noJ. KGJ. z, <btnt: x5 + 4Y = 2013z . Ai>GTJ. llapa'tT)poUJJ£ 6n o apt9J.16� 2013 Statpei-rat J..l£ -ro 1 1 , evro o eivat taot}1t6A.omo� J..l£ -rou� 0, ±1 modulo1 1 . E1tet5ft o 4Y eivat taot}1t6A.omo� J..l£ -rou� k e {1, 3, 4, 5, 9} modulo 1 1 , 1tpoJC61t'tEt

x,y

6n 11 J.10Va5tKft Suva'tft 1tEpi1t-reomt eivat va txouJJ£ x5 = -1(mod l l) Kat

4Y = 1 (mod l l) .

EYKAEI!IIl: B' 89 ""r. l/20

------

Ma&qpanKo( 4wyOMCJpo( - Ma&qpan� OA.'Upm.M�

------

teA.wtaia aA119ei>et av, Kat J.l6vOV av, S y , o1t6te 11 SeSoJ.Ltvrl e�(acoaTJ e(vat tfl� J.lOp<pi]� y

61tou a = X Kat b = 45 . eixaJ.LB 6tt

as + bs

= 2013z = 3z · l l z · 61z '

(1)

2 x ,a<poU Sta<popenKa a1t6 2 1 2023 , ato1to. Apa 1tpe1tet ( a, b ) = 1 . Em1tA.tov txouJ.LB as + bs = ( a + b ) ( a4 - a 3 b + a2 b2 - ab3 + b4 Kat I11J.lttOOvOUJ.lB 6tt

tflV aPXUdJ axtaTJ 9a

)

a4 - a3b + a2b2 - ab3 + b4 = Sa 4 - (a4 + a 3b) + ( a2b2 - a4 ) + ( a2b2 - a4 ) - (ab 3 + a4 ) + (b4 - a 4 )

= S a4 + ( a + b)P(a, b) 61tou P(a, b) 1tOI.:urowJ.lo tcov a, b J.lB aKtpatou� cruvreA.em�. E1toJ.ttvco�

( a + b, a4 - a3 b + a2b2 - ab3 + b4 ) = ( a + b, Sa4 )

(a,b)=l

( a + b, S ) = 1,

J.lB tflV teA.eutaia ta6tflta va tax6et St6tt a1t6 tflV ( 1) 1tpoK61ttet 6tt o a + b 1tou dvat Statpetfl� tou as + bs = 20 1 3z = 3z · 1 1 z · 61 z , Sev J.l1tOpd va txet eo� Statpetfl tov S •

Itfl c:ruv£xeta 9a a1toSe�ouJ.lB 6tt 3 a4 - a3 b + a2b2 - ab 3 + b4 • llpayJ.tan:

Av o

Statpooae tov apt9J.l6 a4 - a3 b + a2b2 - ab 3 + b4 , t6te

3 ab (Sta<popenKa 9a eixaJ.lB Kat 3 1 b , tlt01to Ka9oo� ( a, b) = 1 ). 3k ± 1 ' 1tOU c:ruve1tayetat 6tt Kat 0

3 1 a it 3 1 b , tatco 3 I a , o1t6te 11 (1) 9a tStve 1tco� E1toJ.ttvco� 3 Ia Kat 3 Ib ' Kat apa Ot a, b dvat tfl� J.lOp<pi]� ab ( a2 + b2 ) dvat tfl� iSta J.lOp<pi]� 01t6te 3 Iab ( a2 + b2 • OJ.lco� a1t6 tflV ill11 , Ka9oo� ot a, b eivat tfl� J.lOp<pi]� 3k ± 1 txouJ.lB 6tt o 3 Statpd tov a4 + a2 b2 + b4 Kat a<poo aPXtKa 'U1to9taaJ.lB 3 1 a4 - a3b + a2b2 - ab3 + b4 txouJ.LB 1tco� 3 1 ( a4 + a2b2 + b4 )::_( a4 - a 3 b + a2b2 - ab3 + b4 = ab a2 + b2 , at01tO. 6tt

)

E1toJ.ttvco�, 1tptlyJ.latt tax6et 6tt

3z I as + bs = (a + b)

(

a4 - a 3 b + a2b2 - ab3 + b4

)

a+b ,

(

)

Kat e1tet5ft A6yco tfl� (1) txouJ.LB

e1te'Tat 6tt

a4 - a 3 b + a2b2 - ab 3 + b4 ,

a4 - a3 b + a2b2 - ab 3 + b4

)

3z I a + b .

T6te,

Ka9oo� dvat

Kat a<poo ot a4 - a 3 b + a2b2 - ab3 + b4 , a + b dvat 1tprotot

J.lBta�u tou� J.lB ytV6J.lBVO (a1t6 tflV ( 1 )) 3z · 1 1z · 61z , 9a 1tpe1tet va eivat 61z = a4 - a3b + a2b2 - ab3 + b4 Kat 3z . 1 1z = a + b , Eite 61z ·l lz = a4 - a3b + a2b2 - ab3 + b4 Kat 3z = a + b . H 5eUtBPfl 1tBpi1ttCOaTJ a1toppi1ttetat St6tt ( a + b)4

a4 - a3 b + a2b2 - ab3 + b4 ::::)

( 3 z t 61z .u z ::::) 9z ·9z 6 1z ·l l z ' tlt01t0.

H 1tPOOtfl 1tBpi1ttcoaTJ e1tiaTJ� a1toppi1ttetat St6tt t6te

( t ( r

( a + b )2 = 33z = 332

�

61z = a4 - a3b + a2b2 - ab3 + b4 ,

BVOO 1t.X. 'U1t09BtOVt� XCOP� �Ml�11 tfl� yevtK6tflta� 1tCO� a � b BXOUJ.lB

a4 - a 3 b + a2b2 - ab 3 + b4 = a 3 ( a - b) +a2b2 ( a - b) + b4 � a3 + ab2 + b4

� 2a2 + ab + b2 � a2 + 2 ab + b2 � ( a + b)2

at01to. E1toJ.ttvco� 11 SeSoJ.Ltvrl e�(acoaTJ Sev txet KaJ.l(a AUat'J atou� 9enKoo� aKtpatou�. 2 o.; -rpo1roc;. A1t6 tfl axtaTJ (1) Kat ( a, b ) = 1 CJUJ.l1tepaivouJ.lB 6tt a'Utft J.l1tOpd va ypa<pd CJtfl J.lOp<pi] as + bs

A · B 201 3z , =

61tou

A = a + b, B

)

a4 - a3 b + a2b2 - ab 3 + b4 •

(

E1tt1tA.tov, a1t6 tflV

B = A ( a3 - 2a2 b + 3ab2 - 4b3 + Sb4 , £1tetat 6tt (A, B) = A, Sb4 S . E1tetSft 6J.lco� o S Statpd to 20 1 3Z , e1tetat 6tt (A, B) = 1 . E1toJ.ttvco� 9a txouJ.LB A uz Kat B = vz , 61tou

ta6tflta Sev

EYKAEI411l: B' 89 '1'.1/21

------u, v 9ttu<.:o{

Mu8qpuT1.Ko( A1.uymv1.crpot - Mu8qpunKt� 0).t>pmci6�

a.Ktpa.tot tstotot roc:nt 1 4 4 4

uv = 20 1 3

------

. OJ.lro� a.1t6 tT)V a.vtcr6tT)ta. trov J.lScrrov, 1tpo1C'(mtEt

1 s 4 => - u � v � u => - u � uv =

20 1 3 � u s => 5 � u � 8 , to 01toio tiva.t 16 16 16 a.ouva.to , a.<pou to 20 1 3 oev txst ota.tpsts� a.1t6 to <JUvoA.o { 5, 6 , 7, 8} . 6n

-A �

B�A

TipoPA.1JJ.ta 3. 'Ecrrm

To m>vo#.o Tmv 9HtKrov . npuytuiT1.Kmv up1.8prov. Nu pp.:9oi>v 6u� 01. auvup-rqot:� f : S3 � S IDctrE, "flU 6#.ot>� TOt>� 9£1'\KOi>� 1tpU"(pUT1.KOi>� Up19poi>� X, y, z KU\ k VU IKUV01t01.0i>VTU1. 01. UK6#.ot>9E� 'Tpt:� auv91lKt:�: x, ky, k 2 z = lif (x, y, z ), (y) f (l, k, k + 1 ) = k + (a.) xf (x, y, z ) = zf (z, y, x ), ( p)

)

Ai>GT).

Xp1JmJ.l01totrovta.� to J.lEtO.<JXTIJ.lO.ncrJ.l6 g ( x, y, z) = � f ( x, y, z) ot crov91;1CE� tou 1tpopA.iJJ.la.to� y k 1 . y{vovta.t: (a.') g (x, y , z) = g (z, y, x) , (P ') g ( x, ky , k 2 z ) = g (x, y, z) , (y ') g ( 1 , k, k + 1 ) = A1t6 t� ouo 1tprotE� crov9iJ1CE� A.a.J.lP6.vouJ.LE:

(

g (x, y, z) = g x , on6te,

a:v

oplaou !J£

'l1l

!/;• ) I

( }; ) + i).

g l,

crov<iptTlGTJ h ( w ) = g ( I , I , w ) , t6te g ( x, y, z) = h

crov6.ptT)aTJ h t1Ca.vo1totd tT)V crov9T)1CTl h '

(7) k+l

k+1

;

(j}

67tou �

k+1 1 + �1 + 4 w tEA.Euta.{a. <JXSaTJ , a.v 9tcrOUJ.lE - = w <=> k2w - k - 1 = 0 <=> k = , oivtt k2 2w 1+ k+1 .Ji+'4'; . h ( w) = = wk = k 2 k+1 H <JXtaTJ O.UtT)' lCJXUEl ' "flU 6A.a. tO. W 1tOU J.l1tOpOUV ' VO. ypa.<pOUV ' <JtT) J.lOp<pT)' W = 7 1CO.l O.Ut6 H

en>J.lPa.ivEt yta. 6A.ou� tOU� 9tn1COU� 1tpO."(J.lO.tl1COU� a.pt9J.lOU� , 61tro� <pa.{vtta.t a.v 9scrOUJ.lE

1 + �1 + 4 w . ME a.vtic:rtpO<pT) a.vnKa.t6.c:rta.aJ1, A.a.J.lpavouJ.lE f ( x, y, z) = y + 1l + 4xz . 2w 2x 'Etcrt txouJ.LE ppst J.lta. to 1toA.u crov6.ptT)aTJ 1tou t1Ca.vo1totd t� �T)tOUJ.lEVE� crov9iJ1CE�, 61tro� 1tpOKU1ttEt 0.1t6 <JXEn1C6 fJ..£:r/.o 1CO.t E1ta.A.TJ9EUaTJ trov crov91')1CroV (a.) , (p) 1CO.t ("() . k=

2°<;

Tp01tO<;. A1t6 n� crov91;1CE� (a.) , (p) Kat (y) , yta. K6.9t x, y, z, a, b > 0 , 1tpo1CU1tttt 6tt:

bz !( a 2x, aby, b2 z ) = bf ( a2 x, ay, z ) = b · + · !( z, ay, a2 x ) = f ( z, y, x) = E. f ( x, y , z ) ax a E1t1MyOUJ.lE tou� a.pt9J.lOU� a, b > O stcrt roc:rtE ( a2 x, aby , b2 z ) = ( 1 , k , k + 1 ) , yta. 1Ca1toto k . Av y ( y + �l + 4xz ) 1 y + �l + 4:XZ �

1t6.pouJ.LE a 2 x = 1 Ka.t b 2 z - aby = 1 , OTJMOTJ, a.v: a =

, b=

EYKAEIAim B' 89 T.l/22

2zvr x

2xz

------ MuDqpunKot AuryiDvurpoi - MuDqpunKtc; Ol.:upnulo�

t6n: t\>1coA.a xporimn:t 6n

------

a a a f( x, y, z) = - f(a2x, aby, b2z) = - f ( 1,k,k+ 1) = - ( k+ 1) b b b

y +�/ +4xz �

EuKoA.a t1taA.119£'6ouf!£ 6tt 11 crovciptll<:rll auti) tKavoxotti n<; crov9i)Kt<; tou xpo�A.i)f.Lato<;.

flpoiJA.llf.la 4. �E tva f.1«811f.laTtKo otayrovurpo Kcixotot otayrov�Of.lEVOt Eivat <pwn. H <pi.A.(a Eivat xciVTOTE Uf.lotPaia, Ot)A.aol], oTav o A Eivat <piA.o<; TO'U B , TOTE Kat o B Eivat <piA.o<; TO'U A @a A.tf.lE OTt Ot ota<popETtKOi f.lETa;i> T01><; otayrov�O)IEVOt Ap � , . . . , A, ' yta n

�3,

•

op�o1>v tva aaOsvtiJ� tplllKO KVKlo, av o

1, 2, . . . , n

OEV Eivat <piA.o<; f.lE TOV

A;+1 ,

yta Kci8E

(A,+I = � ) ' Kat OEV 1>1tUPX01>V alla �Ei>'Yll f.lll <p(A.(l)v f.lETa;i> T(l)V otayrovt�Of.1Ev(l)V

TO'U Ki>KAo'U a'UTOi>. IKavoxotdTttt E7nxA.tov 1) aKoA.o1>81) <n>v81\K11 : 11a Kti.Os dla'fWVl(op.svo C , Kti.Os aaOsvtiJ� tplllKo� KVKlo� nov t5sv nspllap.pti.vst Tov

nspllap.pti.vsl TO xo).v tva Jl'l tp{).o TOV

Na (J.1t00EtX8Ei OTt oA.ot 01. otayrovt�Of.lEVOt GE (J.'UTOV TO f.lU81))1aTtKO OtayrovtGf.lO Eivat O'UV(J.TOV va T01t08ET1)8oi>v GE Tpia 0 (1))1QTta, tTGt OlGTE GE KU8E 0(1))1QTtO oA.ot va dvat f.lET«;i> T01><; cpiA.ot. •

Ai>Gll. E>tropouf!£ to 'YPci<p11f.La G (o11A.aoi) tva O'l)voA.o <:rll f.ltirov crto txixtoo f.La�i f.l£ tva O'l)voA.o tu9u"(paf.Lf.100V tf.l11f.lcltrov f!Eta�u Kaxotrov ax6 ta <:rll f.ltia autci) tou oxoiou ot Kopucp€<; tivat ot ota"(rovt�6f.L£Vot, 6xou uxapxtt f.lia xA.tupa avaf!Ecra crt ouo ota"(rovt�6f.Ltvou<;, av Kat f.L6vov av autoi otv tivat cpiA.ot. T6tt t<Jx;6tt: Al\f.lf.la. Ynapxez pia Kopvrp� 'Ul' G onoia, o Pa8p6, (t51'/).at5� ro n).�Oo, rcov fJlt'/Jlarcov aro ypatpt'/Jla Jle eva aKpo 'U7V Kopvrp� av�) eivaz ro no).6 2. Axoott;ll. E>a axoot�OUf.l£ to �11tOUf.l£VO uxo9€tovta<; 6n Kci9t Kopucpi) txtt �a9f.L6 tOUAclXtcrtOV tpia Kat KataA.i)"(OVta<; crt clt01t0. 9tropOUf.l£ tO f.LaKpUttpo £1ta"(6f.l£VO f.LOV01tcltl P = u0u1u2 ...uk tou "(pa<pi)f.lato<;, 011A.aoi) to f.LOVoxcin crto oxoio otv uxcipxouv ouo f.l11 "(tttovtKE<;

lCOpU<J>E<; 1tOU va crovotovtat f.l£ tva tf.liJf.La. Acpou 11 KOpuq>i) Uo EX£t �a9f.L6 touA.cixtcrtov 3, 1tpE1t£t Va <n>VOEttat f.l£ ciA.A.£<; OUO KOpU<pE<; V Kat W , Ot 01tOi£<; avi)KOUV £Kt6<; tOU f.LOV01tattOU P . Exttoi) to P tivat to f.LaKputtpo t1ta"(6f.Ltvo f.10voxcin, ot Kopucpt<; v Kat w 9a txouv "(tttovtKs<; Kopuq>t<; crt aut6 (otacpoptttKa to vu0u1u2 ...uk i) to wu0u1u 2 . . .uk 9a i)tav tva f.LaKputtpo £1ta"(6f.l£VO f.LOV01tcltl, clt01t0 ). 'Ecrtro

Kat

Ol "(tltOVtKE<; KOpUq>E<; crto P trov

Kat

avticrtotxa, f.l£ tOU<; f.ltKp6ttpou<; ouvatou<; OtilCtt<; i Kat j Xropi<; �A.ci�11 tll <; "(£VtK6tllta<;, uxo9ttouf!£ 6tt i � j . T6tt ot Kopu<pt<; u0 , uw .. , uP v crmf.Lati�ouv tvav acr9tvro<; cptA.tK6 ICUICA.o, .

crtov oxoio to

EXtt ouo "(tttovtKE<; Kopuq>€<;, tt<;

Kat

uj , citoxo.

It11 crovfxtta 9a axooti�OUf.l£ to �11tOUf.l£VO f!£ t1ta"(ro"(i) xcivro crtov apt9f.l6 trov xA.tuprov tou "(paq>i)f.Lato<; G . fta n � 3 , to �11tOUf.l£VO tivat xpocpavt<;. 'Ecrtro 6tt tcrx;Utt to �11tOUJ.l£VO "(ta n - 1 . Iuf.Lq>rova f.l£ to ATJf.lf.La, uxapxtt Kopuq>i) v tou G �a9f.LOU to xoA.u 2. A<; xapatllpiJcrouf!£ xro<; to 'YPcl<l>11f.La G' xou xpocixttt ax6 to G f.l£ acpaipt<:rll tll<; Kopu<pi)<; v Kat 6A.rov trov 1tAtupcOV 1tOU EXOUV Eva clKpO tllV KOpU<J>TJ V , tKaV01t0t£i tt<; crov9i)Kt<; tOU 1tpO�ATJf.lato<;, 01t6tt Ot Kopuq>t<; tou f.L1topouv va otaf.Ltptcrtouv crt tpia Orof.Lana, stm c.Ocrtt crt Kci9f: Orof.Lcino 6A.ot va tivat f.Ltta�u tou<; q>iA.ot. Exttoi) 11 Kopuq>i) v sxtt �a9f.L6 2 crto G , o11A.aoi) uxcipxouv 2 to xoA.u tf.liJf.Lata tOU G f.l£ aKpo tllV V ' 11 V Otv crovosttat f.l£ tf.liJf.La tOU G f.l£ tva touA.cixtcrtov ax6 ta xapaxavro tpia Orof.Latta. Toxo9ttrovta<; tllV v crt aut6 to Orof.Lcitto, ta tpia autci Orof.Latta xMov Otaf!Epi�ouv tt<; Kopuq>t<; tOU G rocrt£ crt Ka9tva tou<; va �picrKOvtat f.LOVO cpiA.ot, 01t0t£ to 1tp6�A11f.La f.La<; aA119tU£t Kat "(la tov apt9f.l6 n . EYKAEIAIIl: B' 89 T.l/23

------- Ma&r)panKoi AuryrovUJpoi - Ma&r)panK� OJ..vpm«a�

-------

AaK�ae:tc; yta dtaywvtaJ.louc; Em'Tpo� �uxy(l)vu:rpcbv 'fll � E.M.E.

1\uaEl� 'tEUXOU� 88 KUl -ra CJ'I)pda c' D -roo t:V6� £K -rcov 6i>o 1Jflt.K1>Klicov. Ot. t:q»am6p£Vt:� -rov K'6Kloo <JT« CJ'I)pt:ia C, D Ttpvov-rat. <JTO E . Ta E1>9i>ypuppa 'Tfl'll fi«'T« AD Kat. BC Ttpvov-rat. <JTO F , EVID ot. E'U9� EF Kat AB -rtpvov-rat. <JTo M . Na axo6t:f;E'Tt: 6-rt. 'T« CJ'I)pt:ia E, C, M Kat D dvcn opoK1>Klt.Ka. [Kfva WMO 2010] A1lm](l0� 'Tp01t0�) �tpOUJ.lE Ta eu&UypaJ.lJ.la TJ.liJJ.laTa EO, CO, DO, CA Kat CD . 'ExouJ.tE

r13. l:.t: K'6KJ.o Kmpoo 0 9t:copo6pt: 6u1pt:-rpo AB

�

L.COE = LCOD = LCAD = LCAF , CE CO = . E1tEwi) Eivat CF CA LECF = go· - LBCO = LOCA Kat Ta Tpiyrova ECF, OCA eivat 6J.10ta, 01t6TE LCFE = LCAO . Enewi) eivat LBFM = LCFE LFBM + LBFM = LFBM + LCAO = go· , . tneTat 6n a<pou L.ACB = go· . EnoJ.ttvro� txouJ.tE 6n EM l. AB , on6te Ta CJT)J.tEia O, M, D, E, C Eivat on6te Ta Tpiyrova

OJ.lOKUKAtKcl. · 'a '

COE

Kat

COE

Eivat 6J.1ota Kat

2°� 'Tpoxo� �tpouJ.lE

�

OC,

OD mt OE.

E1tEwi)

TO TEtpanA.eupo ECOD Eivat eyypa'JitJlO. 'ETat, av 0 = M , Ta CJT)J!Eia E, C, M Kat D eivat OJ.lOKUKAtKa. 'Eatro t<'opa 6n O ;e M Kat taTro G = BD rl AC Kat H1 = GF rl AB. 'Eatro

LOCE+ L.ElXJ =9(1 +9(1 = 180',

To ixvo<; 't'I'J� Ka9e't'l')� an6 To

npo� 't'I'JV

AB . 'ExouJ.tE

T6te AC l. BG, BD 1. AG , 01t6te to F Eivat op96Kevtpo Tou tpty<'ovou ABG . EnoJ.ttvro<; Eivat Kat GH2 1. AB . __

EmCJ11 � txouJ.tE

LCH2D = LCBF + LDAF = t so· - 2LBGA LCOD = 1 80. - LBOC - L.AOD = t so· - ( t so· - 2LGBA ) - ( t so· - 2LGAB ) = 2 (LGBA + LGAB) - l SO• = 1 80. - 2LBGA, on6te Eivat LCH2D = LCOD , 5TJA.a5i) Ta CJT)J.tEia C, O, H2 Kat D eivat OJ.lOKUKAtKa. Apa txouJ.tE LECO = LEDO = LEH10 = go· , on6te ta CJT)J.lEia E, C, O, H Kat D eivat OJ.lOKUKAtKa. EnoJ.ttvro<; H1 H2 , 5TJA.a5i) EF 1. AB Kat TtJ.lVOVTat ato CJT)J.lEtO M , 01t6te ta CJT)J.tEia E, C, M Kat D eivat OJ.lOKUKAtKa. =

EYKAEIAm

B' 89 T.l/24

------

A19. 'Emro

Ma9qpaTtKoi Atay(l)vuJpoi - MaOqpaTtK� Ol.:upnwo�

ap a2 , , an , b1 , b2 , , bn • • •

• • •

aK6M)1)6E� 10t6T1)TE�: (1)'

:L(a; + b; } = l,

-------

pl) apvt)TtKoi npaypaTtKoi apt6poi 1t01> tKavonotouv � n

(2)

10 , IO + k 2

max { ak , bk } �

(3) L i 2 · (a; + b; ) = 10.

:L i · (a; - b; } = O, yta Ka6E

k = 1, 2, . . . , n . [Kiva WM0 2010]

AVGfl

Ax6 'tt<; OtOOJ,livt<; cruv6i}Kt<; Kat 'tllV avt<JO'tll'ta 'tO'\) Cauchy' exuat O'tt

( ka,

)2 �

(t. ia, J = (t. ib. ) (t. i'b, )(t. b, ) = (10 - t. i'a. )(t - t. a, ) '

�

= (10 - ka 2 ak ) (1 - ak ) = 10 - (IO + e ) ak + k2 ai . ak � 10 2 , yta Kaut k = 1 , 2, . . . , n . 0 Jlotro<; t1tt'tat E1tOJJ£Vro<; ' tXOUJlt ' ' ' tcrX\)tt ' bk � 10 2 , yta on IO + k IO + k Kcl6t k = 1, 2, . . , n , 01tO'tt 1tpOKU1t'ttt Kat 'tO �fl'tOUJ.lf:VO <ruJl1tepacrJla. '

N13. Na npoaotopiaETE 6u� T� O'UV«Tt� Ttpt� T01> aKtpato'U

6EnKoi aKtpatot

Kat

b+1 + a+1 = k a b

TETOtOt romE

AUGfl

L'ta6tp01totOUJ.lf: Jlla ouva'ti} 'ttJlll 'tO\) tKaV01tOtOUV 'tllV t<JO'tll 'ta

k yta Tov onoio 1>1tapxo1>v [Kiva WMO 2010]

Kat aVUJ.lf:<Ja <J'ta �euyupta aKepatrov

(A, B )

1tOU

B+l + A+l =k ' A B t1tt.A.tyOUJle eva �tuyupt (a, b ) 'te'toto rocr'tt 'tO b va ti.vat tMlXt<J'tO. T6'tt 11 t�i.crroa.., x2 + ( 1 - kb ) x + b 2 + b = 0 txtt Jlta aKepata A.ua.., X = a. AV a' ti.vat 11 OtU't£P11 A.ua.., 'tll <; t�i.crroa..,<;, 'tO'tt ax6 'tt<; crxecrtt<; a + a' = kb - I Kat aa' = b ( b + 1 ) , e1tt'tat O'tt a' E z Kat a' > 0 ExoJ,ttvro<; 6a tcrXt}tt O'tt b + l + a' + l = k. a' b Ax6 'tllV ux66ta.., yta 'tO b exouJlt O'tt a � b Kat a' � b . ExoJ,ttvro<;, eva ax6 'ta a Kat a' t<JOU'tat Jlf: b . Xropi<; �Ml�11 'tll <; ytvtKO'tll 't<l<; uxo6e'tOUJle O'tt a = b . TO'tt exouJ.!f: k = 2 + � E z 01tO'te b b 12 , 011Aa0i} ttVat b = 1 i} b = 2 Kat k = 4 � k = 3, avricr'totXa. ExoJ,ttvro<;, ot ouva-re<; nJ.tt<; 'tO\) k eivat 3 yta a = b = 2 , 4 yta a = b = 1 . .

AlO.

'Emro

aKtpato�,

AEJ1E 6Tt 0 apt6p6�

X j

n � 3 . rta T1)

J1ETcl6Eat)

p = (x1 , x2 ,

ppiGKET«l J1ETU�U T<OV apt6prov EYKAEIA.HI: B ' 89 T.l/25

X ;

• • •

, xn

Kat

)

xk ,

{1, 2, . . . , n } i < j < k . rta

TO'U (J1)V6A01> QV lCJXUEl

------ MafhtJUlnKoi Atayrovtopoi - MaOttfJU'ttK� 0A1>J.11tWOt� napaOEt"{Jla, «m] fl&Tcl9EG1J ( 1, 3, 2,4) TO 3 ppi«rKETat flETa;'l) Tprov 1 Kat 4, alla TO 4 OEV

-------

ppi«rKt'tat Jlf:Ta;'6 TO)V

Kat

f1ETa9t«rEt<; Tou (J't)Vol..ou

'E«rTO)

{ 1, 2, . . . , n }

ota<popETtKrov apt9f1IDV T01> (J't)Vol..ou aUcov ()'l)o

(Jf:

Kcl9E fl&Tcl9EG1J

P; E s.

•

s = {pl ' p2 , ... , pm } anOTtUiTat

Yno9tT01>flE

{ 1, 2, . . . , n } ,

tvac;

ano Otaq»>pETtKE<;

oTt Jl&Ta;'6 onmrovo'l]noTt Tpteov

�:;

auTrov OEV ppi«rKETat f1ETa;'6 Trov

Na npo«rotopi«rETE Tl) flEyt«m] Ttfl'l] T01)

{Kiva Girl's MO 2010]

A'6 G1J

apxft, Jl£ sxaycoyft 0)� xpo� 'tO n ' 6a axoosi;ouJJ£ on m ::;; 2 n . rta n = 3 ' av uxo6eaOUJJ£ on 0 3 osv �ptaKS'tat Jl£'ta;'6 'tCOV 1 Kat 2, 'tO'tS sivat ouva-ro va EXOUJJ£

Ka:t'

S = { ( 1, 2,3) , ( 3, 1, 2), ( 2,1,3) , ( 3,2, 1)} , 011A.aoi} m = 4

23 I . -

'Ea-rco on 'tO �11't0UJJ£VO taXt}st yta n = k ;;::: 3 . ea axoosi;ouJJ£ 'tO �11't0UJJ£VO aA116sust Kat yta n = k + 1 . escopOUJJ£ eva aUVOAo Jl£'ta6eascov Sk+I 1tO'U tKaV01tOtSt 't� auv6i}KS� 'tO'U xpo�A.ftJ.Lam�. Ilapa'tf)pOuJJ£ on, av -ro a-rOtxsio

a'UVOAO'U

Sk +I ,

'tO'tS 0t J.LS'ta6eas�

k+1

Otaypa<psi axo Ka6s Jl£'tU6ScrfJ

1tO'U 1tpOKU1t'tOUV axrJJ.La'tt�O'UV eva aUVOAo

P; Sk

-rou

1tO'U

tKavoxowi -r� auv6ftKs� -rou xpo�A.ftJ.Lam� yta n = k . Tropa sivat apKuo va axoosi;ouJJ£ -rov aKOAou6o t<JXUptaJ.LO: Ynapxovv TO 1l'OAV �6o �zacpopemdc; petaOtaelc; p Kal q mo aVVOAO sk +I Ol 01l'Ofec; aVflaTOlXOVV tm]V f�za peta0e(J1] r E

Sk , orav rove; �zaypalfiOVJle fO aTOlXelO

k+1 .

IIpayJ.LU-rt, a� uxo6eaouJJ£ on

PI = ( XI , x2 , ... , xk +I ) , P2 ( YI , y2 , . .. , yk +I ) ' P3 = ( ZI ' z2 ' '" ' zk +I ) E sk +l ' JJ£ qi q2 = q3 = q . A6yco cruJ.LJ.LS-rpia� J.LXOpouJJ£ va uxo6eaouJJ£ on q = ( 1, 2, ... , k) . Yxo6e-rouJ.Ls on X = Yb = zc k + 1 . IIuA.tv, A.Oyco a'UJ.lJlt'tpta�, J.l1tOpOUJJ£ va uxo6eaOUJJ£ on 1 ::;; a < b < c ::;; k + 1 . �11JJ£tOOVO'UJJ£ on, S1tSt0ft qi = q2 = % = q ' 'ta a, b, c sivat Ota<popsnKU. =

escopouJJ£ -rropa -rou� apt6J.Lou� a, b Kat k + 1 . 'ExouJJ£ PI = ( ... � k + 1, a, . . . , b, . . . ) (-ro a �ptaKS'tat JJ£-r� nov

k + 1 Kat b ) p2 = ( ... , a, ... , k + 1, b, . . . ) (to k + 1 �piaKs-rat JJ£-ra;u -rcov a Kat b ) p3 ( . . . . . . , b, ...k + 1, ... ) (-ro b �piaKS'tat J.LS't�u -rcov a Kat k + 1 ) =

, a,

ExoJ!Evco�, Ka6eva� axo -rou� apt6J.LOU� a, b Kat k + 1 J3piaKuat aVUJltaa aou� UAA.ou� ouo as KU1tOtS� J.LS'ta6east� 'tO'\) sk ' xapa�tU�OV'ta� E'tat 'tO'\)� xsptoptaJ.LOU� 'tO'U sk . ExoJ!Evco� 11 uxo6saft J.La� ft-rav A.av6aaJ.LEvll , oxo-rs -ro xoA.u ouo a-rOtxsia a-ro as eva a'tOtXStO 'tO'U

Sk+I

J.L1topouv va axstKovta-rouv

Sk .

Mevst va axoos�O'UJJ£ on 11 nJ.Lft Jl£'tU6Saf) p sxaycoytKU.

m = 2 n -I

J.l1t0pst va sm-rsux6si. ea Ka'taaKSUUaOUJ.lS 'tf)

EYKAEIA� B ' 89 T.l/26

------ Ma91)paT\Koi Alayrovurpoi - Ma91JpanKt� OA.1>pnuJ.IiE�

-------

Bijpa 2° : ME't<i 'tllV 't01to9€'tll crT) 'tOlV apt9J.Lrov 1, 2, . . . , f 't01t09E'tOUJlE 'tOV apt9J.10 f + 1 ei'te apta'tepa ei'te oe�ta a1to oA.ou� 'tou� JlEXPt 'tffipa 't01to9E't11 JlEvOU� apt9J.Lou�. E1tetofl u1tnp:xouv ouo n JlOVO 1tt9avt� 9taet� yta 'tOU� apt9J.10U� 2, ... , n , Jl1tOpOUJlE va Ka'taaKEUUO"OUJ.lE 2 -l 'tE'tOtE� J.LE'ta9taet�. fta Knee 'tpE� apt9J.Lo6� 1 ::::; a < b < c ::::; n , o c oev �piaKE'tat J.LE'ta�u 'trov a, b .

E7tOJlEvOO� au'to 'tO a'6voA.o 'trov

n 2 -l JlE'ta9taerov tKavo1totei � auv9flKE� 'tOU 1tpo�A.flJ.La'to�.

f : JR � � 1tOU tKavo1totouv 'tllV tO"O'tll'ta f ( x + f (y ) ) = f ( x + y ) + 1, yta K<i9E x, y E � .

A20.

Bpei'te oA.e� n� YVTJO"iro� au�ouae� auvap'tflaet�

A2 1 .

dtVE'tat 11 aKoA.oueia

xn ' n E :rf

a1tOOE�E'tE on 11 aKoA.oueia

JlE XI

= a > 0, xn+l = xn +

xn ' n E N*

� n

' yta K<i9E

Eivat YVTJalOO� au�ouaa Kat <ppayJ.tEvl1 7t<ivro.

A22.

Na a1tOOE�E'tE on yta oA.ou� 'tOU� eenKOU� 1tpayJ.LanKOU� apt9J.10U�

N 14.

BpEt'tE oA.ou� 'tOU� <pUatKOU� apt9J.10U�

x- --y--7-) y ( 2 y - z) z ( 2 z - x ) � 1 x ("':-2_ _ . + + y ( 2z + x ) z ( 2x + y ) x ( 2y + z) aKtpatot

Nts. 'Ea<ro s.

p, q

'tE'totot roa'tE

n � 1 . Na

x, y , z

tax(>et<;

n yta 'tOU� 01tOtOU� umip:xouv 1tp0l'tot eenKoi 2 ( n + 2 Y = ( 2n - 1 Y .

�{(:)r:}(3:). . . .(:)}·

n e N ' . Na ano&�· 6n:

(a) U1tnp:xouv U1tEtpot aUV9E'tot <pUatKOi apt9J.10t

't'E't'Otot OOO"'tE 'tO aUVOAO

OEV Eivat

tva 1tA.flpE� aUO"'tll Jla U1tOAOt1tOOV modulo n . ) U1tapxouv U1tEtpot aUV9E'tot <pUatKOi apt9J.10t (p

'tE'totot OOO"'tE 'tO aUVOAO

eivat tva

7tA.flpe� aUO"'tll Jla u1toA.oi1trov modulo

q>( n ) 0 apt9J.10� 'tOOV eenKOOV aKtpatrov J.llKpO'tEprov 'tOU n 1tOU EtVat crxey&:a 1tp0l'tot JlE 'tOV n, 01tOU n eenKO� aKtpato�. Bpei'te oA.a 'ta seUYll eenKrov aKtpatrov (m, ) 1tOU n tKaV01totOUV 'tllV E�tO"OOCIT) 2 + ( n - q} ( n ) - 1 ) ! = n m + 1 . -

N 1 6. 'Ea'tro

rt4. 'Ea'tro M' N Kat p 'ta J.ttaa 'tOOV 1tAEUprov BC, CA Kat AB ' av'tia'tot:Xa, 'tOU 'tpt'}'OOVOU ABC . 'Emro 0 'tO 1tEpiKEV'tpo 'tou 'tptyrovou ABC . Ot 1tEpty�ypaJ.LJ!Evot K(>KA.ot 'trov 'tptyrovrov BOC Kat MNP 'tEJlVOV'tat ae ota<popenK<i crTIJ.LEia X, Y earo'tEptK<i 'tou 'tptyrovou ABC . Na a1tooei�E'tE on L.BAX = L.CAY . rts. 'Ea'tro

AI ' Bl 'ta iXV'l 'tOOV U'lfOOV a1to n� KOpU<pE� A, B ' av'ttO"'tot:Xa, EVO� 'tptyrovou ABC JlE AC * BC . 0 K'6KA.o� k 1tEpv<iet a1to 'ta CIT)J.LEia AI ' B1 Kat E<pU1t'tE'tat 'tll � 1tA.up<i� AB a'to CIT)JlEtO D . Av 'ta 'tpiyrova ADAI Kat BDBI txouv iaa EJ.L�aoa, va a1tOOE�E'tE on: LAIDBI = LACB . EYKAEIAm B' 89 T.l/27

HOMO MATHEMATICUS

H Homo Mathematicus Eivat Jlta crTftA.Tj cno 1t£pt00tK6 �. ll£ crK01t0 TI)V avrallarfl a1tOiji£COV Kat tTJV ava1tTU�T] 1tpo�A.T]JlancrJ.Lou 1tavco cna e9'J� 9&J.Lata: 1 ) Tt Eivat 'ta Ma9T]Jla'ttKa, 2) I1p&1tet i] oxt va OtoocrKov'tat, 3 ) IIotot eivat m KA.aom 'tcov Ma9TjJ.LanKrov Kat 1toto 'tO avnKeiJli:Vo 'tOU Ka9ev6�, 4) IIote� eivat m eqmpJ.Lor&� 'tou�, 5 ) IIo� emcrTij� i] rlaoot l:m<JtTJJlOOV Mat'touv KMTJ yvro(}T] 'tcov Ma9T]JlanKrov yta va Jl1tOp&cret Ka1toto� va 'tou� cr1touMcret. r1a 't'O� cn>V£py«i't'£; ntc; anlA1Jc;: 1tap«iKAT] <JT] ! 'ta Ke{Jli:Va tTJ� crTijAT]� auTij�, CO� 1tp0� 'tO 1tEpteJ(OJ.L£VO 'tOO� Kat CO� 1tpo� 'tO £1ti1teo6 'tOl>�, 9a 1tpE1tf:l va e{vat crl>Jl�t�acna Jlf: 'ta evOtac:p&povra Kat 'tO £1ti1teoo Ka'taVOT]<JT] � a1t6 Jl&pou� 'tCOV 1tatOtOOV.

E1tt1J.EUta: ru1vv11� Kspauapio11�

la. "rt dvat ra Ma01'Jparu((i; " To 1 899 f.!B'tcl an6 epwva crnt; �UO'Ett; Tilt; fECOf.!B'tptat; 0 David Hilbert(1 862- 1 943) napoucriacre to �t�A.io tou Grundlangen der Geometrie. KA.acrcruc6 �t�A.io mo onoio avatpexet Ka8e EpEUVTJnlt; Tilt; fECOf.!Btpiat;. :Eftf.!Bpa, 'tO KOtVO KOJ.1Jlcl't1 O'E 6A.f:t; nt; fECOf.!B'tpiEt; AZyE'tat A1COAV'C1'J Fewf1Grpia Kat Eivat �acrtcrJ.1EVTJ crta aK6A.ou8a a;wJla'ta 6ncot; tf.9r}Kav an6 tov D. Hilbert: : L A�ubpara uvppoA.tjq, II. A�ubpara dlara�ewq, Ill. A�ubpara U70T1'JTaq, IV. A�ubpara uvvexEiaq Tropa, crat; napoucrta�ouf.!£ TllV Etcrayco"ffl tou D. Hilbert crto napanavco KA.acrmK6 epyo tou. «EII:AHlrH. H fECOf.!B'tpia XPEtcl�E'tat (6ncot; H 1tpOKEtJlf:VTJ JlEAETil Eivat Jlt!l vea 1tpocrna8Eta, JlE E�clUou Kat 11 Apt8J.111't1Kft), yta TllV E1t!lKOAOU811 E1t10tC0�11 va K!l'tflO'KEUclO'OUJlE yta T11 fECOf.!B'tpia Ev!l op8ft cruv8Ecrft Tilt;, A.iyet; JlOVO !l1tAEt; Kat nA.ftpst; Kat 00'0 ytVE'tat a1tAO cruO'TilJl!l a�tCOJlcl'tCOV 8sf.!£A.trooEtt; npotacrett;. Ot A.Oyco 8EJ.1EAtroostt; Kat va npocrnoptcr'tOUJlE an' auto nt; nto mtouoaiet; npotacrstg KMOUV'tat �lWfiar:a Tll g fECOJlE'tpiag. yECOJlE'tptKEg npotacrstt; JlE 'tE'toto tp6no, roots 0 O'XJ1Jlfl't10'Jl0t; !lU'tOOV 'tCOV a�tCOJlcl'tCOV Kat 11 E1t11tAEOV 11 O'flJl!lO'tfl 'tCOV Ota<p6pcov OJlclOCOV OtEpsUVTJO'fl Tll g aAA11Aouxiag 'tCOV sivat eva 8EJ.1fl, 'tO (l�lCOJlcl'tCOV Kat 11 �flPUTil't!l 'tCOV O'UJ.11tEpacrJlcl'tCOV, onoio exst E�Etacr8si, an6 TllV enoxft tou EuKA.Eio11, nou npoK1ln'tOuv an6 ta �sxcoptcrta a�troJlata, va O'E 1tclJ.11tOAAEg Kat E�aipstsg J.1!l811Jlfl't1KEg yiVE'tat cra<pffit; <paVEpft» [ltTJrfl: D. Hilbert, «GRUNDLAGEN DER GEOMETRY», a1t6 npayJlatsiEg. To U1tO'If11 �ltTilJl!l KataA.ftyEt O'Til TI)V yepJ.Lavtl<i] EKOO<JT] (Leipzig, 1930), Jlf:'t«ic:ppa<JT] I:. A.oytKft aVMUO'fl Tll g flV'ttA11'1fitg Jl!lg yta 'tO xropo. IIa1ta061touA.oc;, eKO. TPOXAAIA, A9i]va 1995 ] llpOAE"(OIJ.EVfl.

7�

Ip.

"EV1cJ.dt5eta rewperpia, ay0.1C1'J

U11 J.u: io

flOV "

Ms �UO'fl ttt; nA.&Upeg tptyrovou ABf Kat npog to e�cotsptK6 autou K!l'tflO'KE'Ucl�OUJlf: ruxov'ta op8oyrovta nap/Jla ABL\E, AfZH, Bf0A. Eav A1 E0 HA, B1 L\Z HA, f1 L\Z E0 , 'tOtE ot AAt. BBt. fft 'tEJlVOV'tat crto iOto O'flJlEtO V (yvcocrt6 cog U1'Jpeio Vecten tou ABf) U1'Jpeio Lemoine Tou Tptyrovou :Ee Ka8E tpiycovo ot scrcotsptKeg O"UJ.1J.1EtpootaJ.1Ecroi tou o1EpxoV'tat an6 to iOto O'flJlEio L, nou to ovoJla�oUJlE U1'Jpeio Lemoine tou tptyrovou •

Vecten Tptyrovou

•

do Vecten

do Lemoine

p KQ!

Tpia OIJIIEiO KOTOOKtOO/;OVTOl lit Tp6TTO: lit TTpojroAq TOU

TT]V Ota't'61tCO<JT] Kat a1t6oe�'l Jl£ op9oyrovta Kat O:xt ll£ 'te'tpaycova, tTJV Jti]paJ.L£ a1t6 'tov Krocna N. B ' na

<JIJIIIK!pO!JOIIiJJE OI !OU ABr A

OliiJdo Lemoile

AA.BE.rz

Ta Z,H.e Eivar OIJIJtla avdAoya

U11Jldfl avaA.oya :E11J.1Et!l avclA.oya tOU OOO'JlEvOU O'Xll Jlfl'tOg, OVOJ.lcl�OV'tflt EKEtVU ta O'flJlEtfl 1tOU a) KataO"Kwa�oV'tat JlE ng iotsg otaotKacriEg Kat �) txouv ttg iotsg 10t6TiltEg. n.x. m npo�oMg, O'flJlEiou tou nepiKUKA.ou Tll g nA.&Upf.g tou eyyeypaJlJ.lEvou tptyrovou (w8sia Simson). e

EYKAEIAH:E

B' 89 -r.l/28

------

HOMO MATHEMATICUS

-------

Avti9ETO., TO. Tpia. OT)J..LEia. nou opi�ouv 'tT]V eueeia. Euler oev Eivat a.vfi)..oya., y1a.Ti Ka.Ta.crKeua�ovra.1 JlE 01a.q>openK6 'tp01t0 K0.1 exouv 010.q>OpE't1KEc; 1010'tT]TEc;. Ha. «A vro ro �ipare;»

flowc; Givw o anoaroAtac; rrtc; napaKarw E:mmoA.�c;; «EmmoA.fl Tou . . . . . . . . . . . . . . . . . . . . . npoc; Tov Ilp6eopo 'tT]c; AKO.OlJJ..Lia.c; En1cr'tTJJ..LIDV Ila.ptcrirov. Kupw Ilp6eope, TOAJlro va. eA.ni�ro 6n 01. KK Lacroix Kat Poisson OEV ea. eeropflcrouv KO.KO va. 'touc; U1tEV8uJ..Licrro EvO. cr6yypO.JlJlO. <JXE't1KO JlE 'tT] eeropia. e�tcrrocrerov nou Touc; a.va.TeeT)KE eoro Kat Tpetc; JltlVEc;. 01 tpeuvec; nou neptA.a.J..L�UVE1 a.ut6 TO cr6yypa.J..LJ..LO. O.vtlKOUV <J'tT]V Epya.cria. 1t0U ElXO. U1tO�UAzt 1tEpUcrt crto Ota.yrovtcrJ..1.6 y1a. TO J..LEyaA.o �pa.�eio trov Ma.eT)JlO.ttKrov, 6nou towa., cre 6A.Ec; ne; nept1t'trocre1c;, touc; Ka.v6vec; yta. va. a.va.yvroptcreei a.v J..Lia. e�icrroOT) flta.v lJ OXt E1ttAUcr1.JllJ JlEcrro tEtpa.yrovtKrov pt�rov. Dnroc; a.ut6 to np6�AT)JlO. na.poucrtacr'tT]KE roe; Tropa., a.v oxt a.Mva.mv' 'tOUAUXtcrTOV noM OUO'KOAo crtouc; yeroJ..I.Etpec;, lJ E�E'ta.crnKfl enuponfl eKptve EK TQN IIPOTEPO.N 6T1 oev Jl1tOpoucra. va. eixa. Mcret a.ut6 TO np6�AT)JlO., nprotov, 016n MyoJ..Lat . . . . . . . . . , Kat .

Ill.

emnMov Ot6n lJJlOUV q>ot'tT]Tftc; · Kat Jl' EVT)JlEprocra.v 6n xaeT)KE to cr6yypO.JlJlU JlOU. Auto tO JlUelJJlO. ea. enpene va. JlOU flTa.v a.pKet6. DJ..Lroc;, crUJ..L«prova. JlE 'tTJ yvroJ..LT) ev6c; a.�16nJ..Lou J..LeA.ouc; 'tT]c; AKa.OT)J..Lia.c;, EypO.'If O. Kat 1tUA1 eva. J..I.Epoc; tOU cruyypUJlJlO.t6c; JlOU Kat cra.c; to na.poucria.cra.. BMnete, Kl>pw Ilp6eope, 6n ot epeuvtc; JlOU exouv unomei JlEXP1 crtl J..I.EP O. nepinou 'tTJV iota. J..LOipa. JlE eKEivec; nou A.uvouv tov tE'tpa.yrovtcrJ..LO tOU KUKAOU. a.va.Aoyia. ea. t<JX1)cret troc; TO TeA.oc;; 86A.Ete, Ki>pw Ilp6eope, va. JlE a.na.AA.a�ETE a.n6 tT)V O.VT)cruxia., Ka.Avta.c; touc; KK Lacroix Kat Poisson va. OlJA.rocrouv a.v txa.cra.v to cr6yypa.J..LJ..LU JlOU t1 a.v txouv 'tTJV np6ee011 va. Kavouv a.va.cpopa cr'tTJV AKa.OT)J..Lia.. L\exeehe, Kl>pw Ilp6eope, Ta. cre�lJ tou euA.a.�ouc; U1tT)PE'tTJ cra.c;,

1831

Ma.pTiou » {1'f anCJ.VTYfUf/ OTO 'l"BAO� 'f1'/� O"CiJAYf�}

«Ol avvep"/aTec; rqc; anjJ.. q c; "'P0.lfJOVv-epwrm5v»

1 o etfla.. lion: Kava.Te Tllv nproTll Tll� flO.elJ flO.TlKTJ GKE'JilJ ; (ypa«pet o rtropyo� Ka.pou�aKlJ<;} [foto: Brian Buttetworth]

IlpoM:yof1EVa. An6 To cpiAo 'tT]c; crriJA.tJc; L\fJJllJTPTI Kup1a.K6 (AefJva.-Aaptcra.) A.a�a.J..Le To na.pa.Kano KetJlEVO 'tOU ru:.Opyou Ka.pou�UKT). Tov euxa.p1cr'tO'UJlE. «l:e T)A1Kia. Tecrcraprov ETrov ; Tp1rov ; L\uo ; Mnopei va. tl'tO.V �OJllJ vrophepa.. l:'tT]V npa.yJ..LO.'ttKO'tT]'tO., EXE'tE yEVVT)eei Jl0.8T)JlO.'t1K6c; . . . Ta. TEA.eumia. XPOV1a. 01 epEUVT)Tec; Ka.Taq>epa.v va. Jlaeouv 't1 cruJl�O.tVE1 cr'to JlUO.AO 'tOOV 1tOAU Jl1Kprov na.101rov, a.K6J..La. Kat Trov �pecprov. Me nowuc; Tp6nouc; ; 'Eva.c; a.n' a.uTouc; eiva.1 lJ npocrEKT1Ktl na.pa.KoA.oueT)OT) 'tOU �AEJlJlO.'t6c; 'touc;, 'tO onoio, AEvE 01 EpEUVT)TEc;, O'tO.V mpeq>E'tO.t JlO.Kpta JlO.propa Ol.KEtO'tT]'tO. tl 1tAtl�lJ Kat O'tO.V 1tp00T)AOOVE'tat EK1tAT)�T) t1 evota.«ptpov. revtKO'tEpa. 0.1t0 'tT]c; np6crq>a.Tec; epeuvec; Jl1tOpOUJlE va. oouJ..LE notec; eivat ot npocrooKiec; Trov J..Lroprov cre Ota.«popenKec; KO.'tO.<J'tacretc;, Kat nroc; a.vnopouv O'tO.V To O.U'tec; EV'tU1tOlcrtO.K6 na.pa.�ta�OV'tat. <JUJ..L1tEpO.crJlO. 1tOU 1tpOKU1t'tEt eiva.1 on 6A.ot epxoJ..LO.cr'tE cre a.uT6v Tov KOcrJ..Lo tKa.voi yta. 'tTJV KO.TO.VOT)OT) TOOV �a.cr1Krov Ma.elJJlO.'ttKrov.

«rsvv16J..La.crTe JlE EJlq>U'tTJ 'tTJV a.icrST)OT) tou nA.T)eap1eJ..Lou», Met o Brian Butterworth, cruyypa.q>ta.c; 'tOU epyou The Mathematical Brain. «Ta. �ptq>T), O.KOJlT) Kat KO.'tU 'tT]V npro'tTj e�OoJ..LaOa. 'tT]c; �roiJc; muc;, Ka.:ta.voouv noTe a.AA.a�e1 o a.pteJ..Lo c; 'tOOV 1tPO.YJlU'tOOV 1tOU na.pa.'tTJ pouw. 0 iowc; E1tl.OT)JlO.lVE1 0't1 'tO. �pEq>T) EXOUV a.pteJllJ'ttKec; npocrooKiec;. «AV Oei�etc;, O.PX1KU, cre eva. �peq>oc; on KpU�Etc; EvO. O.V't1KetJlEVO nicrro 0.1t0 JltO. oe6VT) KO.t cr'tTJ cruvtxe1a. 6n Kpu�etc; illo tva. a.vnKEiJ..LEVo crTo tOtO JlEpoc;, 'tO Jlrop6 ea. 1tEptJ..I.Evet on mcrro a.n6 'tT]V oe6VT) �picrKOV'tat Mo a.vrtKetJlEVO., Kt a.v a.uTft lJ �E�O.tO'tT]'tU 'tOU O.VO.'tpa.nei ea. vtrocret EK1tAT)K'tO tl 0.1tOYOT)'tEUJlEvO. 'ETcrt, nptv a.K6J..La. Ta. J..Lropa Jl1topecrouv va. ecrnacrouv To �AEJlJlO. Touc; mov K6crJ..Lo , Vtroeouv EK1tAlJ�lJ O'tO.V �AE1t0UV tva. aepotcrJ..LO. JlE A.aeoc; a.no'teAEcrJ..La. ! »

r 9 E f.10.. a.n60El�lJ 'TOU AlJf.lf.lO.'TO�. na.va.ytro'TlJ� OtKOVOf.lUKO�

IlpoM:yof.1eva.. a) crTo npoT)youJ..Levo Te6xoc; Kat EKq>roVT)OT) Tou AtlJlJlO.Toc; olli«pUye, EK na.pa.opoJ..Lfl c;, cru-yKEKp1J..LEva. <J'tT]V ev6TT)Ta. «01 cruvepyaTec; tT)c; tva. A.aeoc;. crrocrriJ a.nooetKTea. eivat T) 1ta.pa.KaTro: crriJ A.tJc; ypacpouv-eproTouw, nproTo eeJ..La., <J'tT]V

EYKAEIAm B ' 89 T.l/29

-----MA

HOMO MATHEMATICUS

Mr MB 1 + -1 + -1 ?.3/2

p) O''tO i010 TeUxO� Kat et�a, 0 <pM� TT)� mltATJ� IlavaylCO'tTJ� OlKovo�aKo� eixe anooe�e1 -ro JC6p1o

--------

et�a, 6�ro�, TJ <i1tOOE�TJ 'tOU 01t0tOU O''tT)pixeTJKE O''tO napaKci'tro A.it��a

TJ EK<j)IDVT)(JTJ Kat U1t00El;TJ 'TOl> Ait ��U'TO�

A iveraz rpiyrovo ABr Kaz M roxaio 01/fleiO evr6' avrov. Oz ev8eie' AM,BM,FM ripvovv Trl' 1rkvpi' BT,AT,AB am 01111eia A J,BJ,ri avriarozxa. Na &zx8ei 6rz

MAl MA + MA 1 + 1 MA

Av x,y,z eiva1 01 anocnacre1� -rou M an6 TT)� nA.eupe� avricr-rmxa, 'TOTE ea 1crx(>ouv 01 lO'O'tTJ'TE�

Br,Ar,AB

( Mar) = ( ABr )

( AMr) = ( ABr )

u 13

( AMB ) = ( ABr )

A1 =

lcrx(>ouv Ot lO'O'tTJ'TE� MA

1_ x AA 1 \)a

(a)

u 13

' BB 1

UJ3

rr 1

?.3/2

MB I MB MB 1

Mr 1

1 A = 2 MA

-- + 1 Mr Mr 1

Mr MB

1 A = 3 MB

�+�+�=1 AI+l /...2 +1 1..3 + 1 1 1 1 = + AI+1 /...2 + 1 + 1..3 + 1 2

Kill UV TT)� 1tpocretcrro KU'tcl �AT) ea f.xro y X Z +-+-=1

-- --

1 + MB 1 + Mr 1

(y)

Enetoit, -rropa, yta Kci9E 'Tptaoa ( UJ,Il2,a3) eenKOOV ap1e�rov tcrx(>et z

(a1 +a2 +a3 )

[

]

(

)

_!_+_!_+_!_ � 9 a1 a2 a3

't� 01t0le� 1tpocref.'tOV'tU� KU'Tcl �TJ, BXOU�E (A.Oyro. ea txou�e 1 1 1 'tTJ� (a)) __ + __+ __ (A.•+1+A2+1+A.3+1) � 9

__1 MA

AA 1

1 + BB 1

X · Y 1 +-+-=1 rr = ua u u .. 'Y 1

.,.,

(p)

AI+} A-2+1 . A3 +1

11 -reA.eu-raia crxtO"TJ, � P nO'TJ 'tTJV (y) ypa<pe-rm

TTJV (p) �1topo6� va 'tTJV ypci'lfOU� OlUOOXlKa

....:1_::

IV. EtOTJ aouA.E�- Etoit aet�

1 '1• The Abel Prize Laureate

Pierre Deligne]

201 3 [foto:

H NopPTJ'YlKlt AKUOT)�ia rpa��cl't(OV Kat EmO"tT)�rov antvet� TO Abel Prize

= 1 it

2 01 3

cr-rov Bf.A.yo �aeTJ�UnK6 Pierre Deligne, -rou Institute for Advanced Study, Princeton, New Jersey, USA, yta "TT) OTJ�loupytKft cru�poA.it -rou O''tT)V aJ..yepptlCft rero�E'tpia "

2" . ZTJTOUV'Tat tpEUVTJ'TE� yta 'TO npoypa��a LR [Lehmer-Ramanujan] [foto:Derrick Henry Lehmer] Na n �a� ypa<pet o NiKo Auyep6� : «Me-rei 'tT)V avaKMU'IfTJ noUOOv np6ypa��a avf.oe�e itOTJ ouo y1yavnou� npro-rou� tOto'Tft-rrov 'TIDY 1tpro-rrov apte�rov 'tOU ap1e�ou� � 1tEptO'O'O'tEpa U1t0 .1 4000 Kill 26000 'I'TJ<pia. l:e O'UVBXElU TT)� OpclO'T)� 1t0U apxtcrE � 'tOY ronou LR, ano<pacricra�e va PaA.ou�e O'E E<pap�oyit 'tO �aeT)�U'tlKO F. Morain emvoitcra� -rou� unoA.oytO"�ou� TT)� 1tp6ypa��a LR 1tOU exet (0� O'TOXO 'tT)V 1tlcrt01tOlTJO'TJ 1tlO''t01tOtT)O'T)�, yta VU ppou� Kat 'tOU� 1tpOO'TOU� �crro 'tTJ� �eooou eA.A.EtmtKrov Ka�1tl>A.rov. Au-r6 -ro apte�ou� Lehmer-Ramanujan (n-ravtKou� npro-rou� EYKAEIAIIl:. B ' 89 T.l/30

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HOMO MATHEMATICUS

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apt9f..lOU<; aufil tTI <popa a<pou �E1tEpvouv -ra xiA.ta f..lE -rou<; apt9f..lOU<; Mersenne. Ee avr6 ro vto 1eA.aimo 'JITJ<pia). To f..la9TJJ.la-ruco np6ypaf.1J.la nEpva n/J:.ov crE JI.Ozpa(ovJ.le apz8J.lof)(; ae O'Vvepyare�. oz o1eoioz f..lia vea q>UO'TJ yta va yivEt f..lia O'UO'tTJJ.lanldt eKTeA.ovv ro 7!p6ypaJ.lJ.la mmo1Coi7J0'1]� PRIMO 1eov mmonoiTJO'TJ o�v -row U1tO'I'TJ<pirov npO:mov XflYfO'lJ.l01COlei 'l'Yf J.lt8o6o ellezmzKwv KaJ.l1CVAWV. apt9f..l6>V LR yta va £xouf.1E evav Ka-ra.A.oyo av6J...oyo [1tTJYI1 : ano tTJV TJAEK'tpovtldt aAATJAoypa<pia J.la<; f..lE -rov N. AuyEpo ]

311 Av EiGTE Kal..oi GTfl 9sropia Trov apl9Jlrov Jl1tOpd va KtpoiGETE 1 EKaTOJlJlUplo ooJ..a p1a ! ! 0 Andrew Beal nap6n Otv EtVat J.laOru.tanK6<; 0 Beal npocr<pepEt 'tO 1t00'0 'IOU 1 EKa't. ooJ..apirov O'E sivat 'U1te'60uvo<; yta J.lta etKa<ria, tva avotKt6 onotov Ka-rop9rocrEt va anooEi�Et tTJV EtKacria i) va 1tp6 P"-11 fla 1tou cpspet to 6voJ.la mu: «Tllv �pEt eva av-rt1tap oEtyf.1a. , , , , , A,ey6f..l£Vll Et.Kaaia BEAL» yvrocrrf] 1tMov Kat Ka-ra f..lta EKOOXll TJ EtKacrta TJTav yvoocrtTJ ano -ro 1 99 Kat ano iOE'tat ? -rov af.1EptKavo f..l a 9TJJ.lanKo : eo<; EtKacria tou 1 EKa-ro upiou ooJ..apioov. Damel Mauldm alla 0 Beat i)-rav aUTO<; 1tOU ((AV Ax + Bv , B, C, x, y Kal z eival "' = Cz, ono "Emldtpu�E" -ro npo�ATJJ!a.

�

:�

8BTIKoi aKipaUJI Kal x, y, z eiva1 oA.01 Tovq peyoJ.vTBpOI ano TO 2, TOTe 01 A, B, c npbcel va izovv iva Ko1vo npwTo napo:yoVTa.J> na napadBI')'Jla t!TYfV IUOTYfTa : 33 + 63 = 35 TO 3 eiva1 KOIVOq napa')'OVTaq TCOV 3, 3, 6 Kal enia,q npwToq aplOpoq .

�

'lcrro<; Otv 1teptP ciAAetat <l1t0 tOV OpuAo tOU teAe'l>taiou Osropi}J.lato<; tou Fermat alli to eivat E1ta0Ao OtOAo'U Oev XP11J.l<lnK6 EUKataq> pOVll t0 . [1tlfY11 : http://www. newsbomb.gr/diethnh/story/3 1 44 1 O/prosferei- 1 ekat-do1aria-se-opoion-apodeixei-to-mathimatiko-provlima]

411 • 'Pf1 Trova [ foto: sir Isaac Newton] To navEmcrfilf..ltO KBf..l1tpt-r�, au-rrov �picrKE-rat TJ npoo-row1tTJ W1tOOf..lBYTJ BKOOO'TJ 01tOU oioaO'KE . 0 OtUO'TJf..lO <; 'tOU aptO''tOUpyi}f..la'tO<; 'tOU . «Principia q>UcrtKo<; Kat J.la9TJJ.lanKo<; Mathematica», J.la�i f..lE tTJ<; Ef..l�OAtf..lE<; Q'XE'tlKB<; IGCXUK Nti>T<OV, E1tt'tpB1tEt XEtp6ypaq>E<; O'TJf..lEtcOO'Et<; Kat anaVtTJnKa crxoA.ta tll <; n/J:.ov yta nprotTJ <popa tTI E1ttKpt-re<; -rou, nou o ioto<; EiXE KavEt navoo <no otKo otaOlKWaKit 1tpocr�aO'TJ 'IOU aV'tt'tU1tO [ypaq>El 0 George onOtouoi)no-rE svotaq>EpOf..lEVOU Koutroulias] (1tTJYil :ano tTJV TJAEK-rpovtldt ma 'JITJ<pt01tOlTJf..lBva XEtp6ypa<pa Kat npoo-rowna allTJA.oypa<pia f..la<; f..lE -rov f..lOUO'tKOf..la9TJJ.lanKo 'tU1tOOf..l£va epya 'IOU f..lE'YUAOU E1ttO'fll f.O.l Va. ME-ra9J NiKO AvopouAaKUKfl]

511• E1tETE\O� 9avaTOU TOU Euler [ foto: Leonhard Euler] Erz� 1 8 EemtJ.lfJPYf 2013 O'VJ.l1CAYfpW87JKav 230 yemJ.lerpia ('l'YfV o1eoia O'VJ.l1CA�pwae Kaz x.p6vza a1e6 rov 8ayaro 'l'Yf� 1JJ.li8eov 'l'Yf� J.la87JJ.larz�� reA.ew1Coi7Jae), O"l'Yf 8ewpia rwv apz8JI.WV K. r.A.. IloUoi J.la87JJ.larzKoi 6poz ((Jipovv . ro 6voJ.la rov, em�JI.Yf�. rov Leonhard Euler. Oz 0'1Cov6az6repe� epyaaie� rov 61ew� Yf ma8epa rov VzA.ep, o ap18poq Tov VIA.ep ava((Jtpovraz O"l'YfV avo.A.v0'1] rwv (To ')'VCOtno e), oz J.leraf3A.7Jrt�. Yf ypaJ.lJ.l� Kaz Yf zao1CepzJ.lerpwv, O"l'Yf O'Vaxtrz0'1] rwv el;iaw0'1] rov Vzkp K. a KVKAlKWv Kaz rwv eK8erzKwv rza 'l'YfV aKpif3eza, ro tpyo rov a1eorekiraz a1e6 75 rOJ.lOV�, O'VvoA.zKa 45000 aeA.i&� Ma87JJ.larzKcbv! O'Vvap�aewv, O"l'Yf 8ewpia 'l'Yf� E1Ci0'1]�. V1Capxovv 4000 xezp6ypa{/Ja (ill7JA.oypa((Jia 1Cepzmpo{/J�� O'WJ.la'l'O� yt)pw a1CO O"l'a8ep6 0'1'/J.leZO, O"l'YfV aVaAV'l'lK� JI.B dla0'1]JI.OV� aVYXflOVOV� 'l'OV J.la87JJ.la'l'lKOV�) . {1C7/Y�: ·

Vikipedia}

[foto: Albert Einstein - Charly Chaplin] Kano-rE cruvavfil9TJKav o Charly Chaplin (o 9puA.tKo<; LapAO>) Kat o Albert Einstein. Kat TOTE: -Albert Einstein: «au-ro 1tou 9auf.1a�oo O'tTJV TBXVTJ cra<;, Eivat 11 1ta-yKOO"f..ltOtTJ'tU tTJ<;. llsv M-rE f..lta IJ:.�TJ Kat Of..lOO <; . . . oA.o<; o KOO"f..lO <; aa<; Ka-raJ..a�aivEt>> --Charly Chaplin «Eivat aJ..i)6Eta a1taV'tast o Chaplin. alla 11 aa<; Eivat aKoJla mo f..lE'Y6J...fl . . . oA.o<; o KOO"f.lO<; aa<; 9auf.1�Et aa<; KataA.a�aivEt. . . . >> [1t1lrfl: avaK'tft9r)Ke an:6 n:eptcrcr6Tepa Tow ouo mails, n:ou scne1Aav <piA.ot 'tll � <m'JI..TJ�l

611 • IlvsvJlaTroos�

oo�a Kat Kaveva<; osv

Ilflo A1tUVTfiGfl GTO <<A VTO TO i;ipaTEj)> A1tomo/J:.a<; tTJ<; Ema-roA.i)<; Eivat o Evariste Galois EYKAEIAm

B' 89 -r.l/3 1

r . a ivav KO I VO TPOTTO iK<ppOOI'JS OTa MaOI'J J.IOT t Ka «

Ms acpopJlit TIJ cppa011 'tou Xpftmou ftawapa : Na txovpe -rq ylwuua m� Aoyuuj Kal -ra MaOqpa-r1Kti. w� ylwuua» Axo TO ruopyo I:. Taaaoxoul..o I:xoA.tK6 l:uJ.LPouA.o Ma61JJ.LanKrov

Ero rdJxor; 65 rov EvKJ..ei�1f B ', iKava pza npomra8eza va e7rUl1Jpavw A.av8aupever; eKrppaueu; Kaz (JI)ppoA.zupovr; nov ave1r.aiu811ra XP1JUZponozovvraz ura Ma81JftarzKa, pe r1J rpzA.o�o�ia va vnap�ez eKrppaurzKf! opozopoprpia ura ap8pa rov 7repzo�zKov. Mera an6 r6ua XP6vza �zaniurwua 6rz dvaz npayparz �vUKoA.o va all�ez o rp6nor; �zaronw(]1Jr; , 6rav an6 r1JV apxf! (rovMxzurov an6 r1JV A ' AvKdov) &v txovpe eurzauez ro ev�zarptpov par; Ur1JV avUr1Jp6r1Jra r1Jr; , �eKa8api(ovrar; ro v611pa nov eKrppa(ez. ea ura8dJ ebdJ, vurepa an6 em8vpia noA.A.wv (JI)Va&A.rpwv , urzr; nA.tov anapair1Jrer; an6 eKEiver; rzr; napar1Jpf!(Jezr;. 1) l:X,tTlKU JlE 'tO aiJVOAO Opla!lOU auvapT1]<fl\�-E�l(Jffi(J1]�

Ka•a 'tTIV ava�it'tll mt •ou auv6A.ou optcrJlOU auvap'tll mt � nolli<; <pop£<; AEJlE 1tpE1tEt Kat EVVOOUJ.lf: apKEi it EVVOOUJlS 1tpE1tEt Kat apKd, EVID aUE� <pop£<; avn9E't00<; AEJ.lf: apKd Kat EVVOOUJ.lf: 1tpE1tEt it EVVOOUJ.lf: 1tpE1tEt Kat apKEi.

np£nst : ( 1 -x � 0 Kat x '* 0) Kat X EVVOOUJ.lf: ' 1tpo<pavro<; 1tpE1tf:t Kat apKEi. AvnA.aJl�aVOJlacr'tE OJlOO<; O'tt Kat aU'tO<; 1t0U ea 1tf:t 1tpE1tf:t X '* 0 ' sivat crrocr't6<;. AnA.ro<; au't6 osv sivat apKE't6. Kanoto<; illo<; Jl1topsi va tCfXUptcr'tEi 6n, yta va op�E'tat 11 f(x) , apKEi x < 0 it tcroouvaJla 6n 11 f(x) op�E'tat 6'tav x < 0 , npayJla aA-118£� JlEV aA.M 6Xt anapai'tll 't O. EnOJlEvOO<; 'tO crrocr't6 sivat va XP11crtJ.I.01totitcroUJlE 'tTIV EK<ppamt : fta va op�E'tat 11 auvap'tTJOT} IlapaouyJla: AEJlE, yta va opi�E'tat 11 auvap'tll mt f(x) =

f(x)= � +

� + l_

� 1tpE1tEt Kat a\fi: ( 1 -x�O Kat x=t;O) it 'tTIV EK<ppamt : H f(x) op�E'tat av Kat JlOVO av ( 1 -

x� 0 Kat x '* 0) it 'tEAo<; 'tTIV EK<ppamt :

H f(x) EXEt m>voA.o opt«JflOV 'tO A= { x e iR/ 1 -x�O Kat x=t;O} =

=(-oo,O)u(O, 1 ] . To iota aKpt�ro<; Jl1topol>Jlf: va nol>Jlf: Kat yta 'tTIV s�icrromt

�

= !X .

Ilpo<pavro<; 11 f(x) opi�E'tat crs Ka9s unocri>voA.o 'tou A, n.x. cr'to A1 = (- oo,O), 6nro<; npoEinaJ.lE, it cr'to

A2= [ _!_ , 1 ) K.A.1t. , OJlOO� OU'tE 'tO A1 ou'tE 'tO A2 sivat cri>voA.o optcrJlOU 'tTJ� f(x).

Ilolli<; <pop£<; av•i va 1tOUJ.lf: 6'tt 11 f(x) fxEt m>vo.A.o opt«JflOO 'tO A = (- oo,O)u (0 , 1 ], AEJ.lf: (KaKro�) 6n 11 f(x) opi�nat cr'to A, oivov'ta<; 'tTJV svronromt 6n 'tO A Eivat anA.ro<; unocri>voA.o 'tou auv6A.ou optcrJlOU 'tll � Kat OXt Ka't' aVUYKTJV 'tO crUVOAo optcrJlOU 'tll <; . Mta anA.it au�it'tll mt cr'tllv •�11, nptv apxicrouJ.lE 'tO KUpiro<; JlUSTJJla, Jl1topsi va npo�A11Jlarlcrst 'tOU<; J.ta911'tE<; rocr'tf: va �EKa9apicrouv Kat va Ka'tavoitcrOUV 'tTJ ota<popa JlE'ta�l), 'tOU 1tpE1tEt Kat 'tOU apKd. A<; �11'tftcrOUJlE A.otn6v an6 'tOU<; J.ta911'tE<; va E1ttOT}Jlavouv nows<; a1r6 n<; napaKa'tro t8t6'tll'tE� nou anooioOUJ.lf: crs 01tOtaoit1tO'tE av9pro1to, t<JXI}ouv. D'tav n.x. AEJ.lf:: fta va Eivat Kanota� an6 'tTIV A9itva 1tpE1tEt va Eivat an6 'tTIV AntKft, EVVOOUJ.lf:: Av Kanoto<; sivat an6 'tTIV A9itva , 'tmE sivat an6 'tTIV AntKft. El>KoA.a otantcr'trovouv m JlaSTJ•£<; 6n au't6 sivat crrocr't6. tnav AEJlE: fta va sivat Kanoto<; an6 'tTIV A'tnKft 1tpf1t£t va sivat an6 'tTIV ASftva , npo<pavro� aU't6 Eivat A.a9o<; ( .::lsv t<JXI}st yta Ka9s av9prono) • Ava.A.oya: fta va Eivat Kanota� an6 'tTIV ASftva apKEi va sivat an6 'tTIV A•nKft, mtJlaivst: Av Kanoto� sivat an6 'tTIV A'tnKft, 'tO'tE sivat an6 'tTIV ASftva, nou npocpavro<; Eivat A.6.9o� ('taU'tOOT}Jl11 Jlf: 'tTJV aJ!Ecrro<; 1tpOTJYOUJlEYTJ) EYKAEIAHl: B' 89 -r.l/32

------ n.a tvav Kotvo Tpono EK«pp«CJ'I'I'ii cna MaOTJJ1«TtK6:

-------

• E�<illou 11 1tp6-ra011 : na va Eivat K6:1totoc; a1t6 'tllV A-r-rtKft apKd va Eivat a1t6 -r11v AefJva Eivat crrocr'tfJ. 'E-rm A.ot1t6v ea avnA.11 cpeouv ot J.La�-rec; 6-rt , 6-rav eeA.ouJ.LE va a1tooEi�ouJ.LE K6:1tota 1tp6-ra011 , oev ea AEJ.LE Tt npintt va ano()�:�oupt aU6: Tt apKd va ano()�:�oupt. Ilapa()dyf.laTa: • na va a1tOOEl�OUJ.I.E 6n DUO -rpiyrova EtVat icra, OEV ea 1tOUJ.lE npi1t£t va exouv iaa Ejl�aoa, Ot6n au-r6 OEV Eivat apKE'tO. E1ti011 c; OEV ea 1tOUJ.LE npi1t£t va exouv -rtc; 'tpEtc; 1tAEUpec; icrEc; jlla 1tpoc; J.Lia, ot6n 'tO'tE exEt KU1totoc; 'tO OtKairoJ.La va aJ.L<pt�O:Aet yta 'tO av Eivat t1 6xt au-r6 apKE'tO, aU6: ea 1tOUJ.lE apKd va exouv -rtc; -rpEt<; 1tA.Eupec; icrE<; Ilia 1tpoc; J.Lia. Av 611ro<; eeA.ouJ.LE va a1tooEi�oUJ.lE 6n ouo -rpiyrova exouv iaa EJ.l�aM J.l1topoUJ.lE va 1tOUJ.I.E apKd va Eivat icra, 1tap ' 6A.o 1tou au-r6 oev Eivat a1tapai't11 'tO (Eivat tcrxup6-rEpo a1t6 -ro s11'tOUJ.lEVO) . • EV'tEMO<; av6:A.oya, OEV ea 1tOUJ.LE 6n yta va ElVat 1tpe1tEt ot6n au-r6 OEV EtVat apKE'tO. 1t.x. ( , evro � 2. M1topouJ.tE OJ.Lro<; va 1tOUJ.I.E 6n yta va Eivat apKEi 1tap' 6A.o 1tou au-r6 oev Eivat a1tapah11-ro (meav6v = • fevtK6: yta va a1tooEi�oUJ.I.E 'tllV aA.fJeEta jltac; 1tp6m011 c; p, �am/;6J.lacr-rE cr-rov Kavova anomuiuewq, crUJ.l<prova J.1.E -rov o1toio oEi:xvouJ.tE apxtK6: 'tllV aA.fJeEta J.Ltac; cruVE1tayroyiJc; J.lE <n>pnipaopa 'tllV 1tp6m011 p 1t.x. 'tll <; PI => p Kat O"'tll cruvexEta 6n 11 u1t6eE011 PI Eivat a1..11 et1 c;. Av OJ.Lroc; a1tOOEi�OUJ.I.E aA.118tl jlta O"UVE1tayroyiJ J.1.E vno9£G1] p, 1t.x. 'tllV p => pz, Kat O"'tll cruvexEta -r11v aA.fJeEta 'tll <; pz , -r6-rE 1tpo<pavroc; oev E�acr<paA.ii;E-rat 11 aA.fJeEta 'tll <; p , acpou u1t6:pxouv aA.11eEi<; O"UVE1tayroyec; jlE 'lfEUOTJ U1toeE011 . E>ujli/;OUJ.I.E on jlta O"UVE1tayroyiJ ElVat aA11etl<; one; 1tEpt1t'tOOO"Etc; (a, a), ('If ,a), ('If, 'If) Kat 'lfEUOTJ<; J.LOVO O"'tllV 1tEpl1t't(l)011 (a, 'If). Ilap6:()styf!a: na KaeE Eivat 1tpo<pavroc; a1..11 et1c; 11 cruvE1tayroyiJ Av OE�OUJ.I.E 6n => Eivat aA.119tl<; 11 U1t69E011 -r6-rE 1tpoKi>1tn:t 6n Eivat aJ..11 et1c; Kat 11 1t.x. yta = {Kav6vac; a1to0"1t6:crEroc;). Av 6J.Lroc; a1tooEi�oUJ.lE 6n Eivat a1..11 eec; -ro O"UJ.l1tepacrjla au-r6 oev Oll J.LaivEt 6n Eivat aA.119tlc; Ka-r' av6:YK11v Kat 11 u1t6eE011 1t.x.

3i>4

x>2

-3

x2 >4, x2 > 4,

x -3).

xeiR

x> 2

(x>2

x>2,

x2>4). x2>4 x 3 x2>4, x>2, (-3)2> 4 .

2) !:J:ETlKcl J1E TTJV avayvro<JTJ TTJ'ij (Jl)VE1tayroyft'ij (p=>q) O"UVE1tayroyiJ (p=>q) Ota�6:1;E-rat (av p, 'tO'tE q), AEjlE OE on 11 1tPO'ta011 p Eivat tKavq cruveiJKll yta 'tllV q, evro 11 1tp6-ra011 q Eivat ava-yKaia cruveiJKll yta 'tllV p. E1ti011 <; AEJ.I.E tcroouv6:J.Lroc; 6n yta va tcrxUEt 11 q apKd va tcrxUEt 11 p iJ 6n yta va tcrx6Et 11 p npintt va tcrxUEt 11 q. IloWc; <pope<; yp6:q>OUJ.I.E (av p=>q) Kat oev avnA.aJ.L�av6J.Lacr-rE 6n au-r6 Oll J.LaivEt (av av p, -r6-rE q) oivov-rac; E1tt1tA.Eov 'tllV EV'tU1tro011 6n -ro " av " ava<pepE-rat O"'tllV 1tp6-ra011 p, evro A.oytK6: avacpepEmt crE oMKA11Ptl Trj cruvE1tayroyiJ (p=>q) . 'E-rm avn va yp6:'1fOUJ.I.E : , yp6:<poUJ.I.E (KaKro<;) (av => • E1ti011 c; KaKro<; XP110"tJ.L01totoi>J.LE 1toWc; <pope<; -ro crUJ.L�oA.o (=>) crav m>ppoJ.o GTEVoypa<pia�. na 1tap6:0Etyjla, av-ri va YPU'IfOUJ.I.E: (( 1tOU Oll J.LatVEt 6n: (( na KaeE X O"'tO tcrxUEt t1 « tcrx6Et 11 O"UVE1tayroyiJ => 4 1tOU Oll J.LaivEt 6n : «rta KaeE x cr-ro => yp6:q>OUJ.LE (KaKroc;): « na KaeE t1 « na KaeE t1 «ME => K.A.1t. => • Ev-rEA.roc; av6:A.oya 6-rav yp6:q>OUJ.I.E av p<::>q {tcroouv6:J.Lroc; av q<::>p , eEropouJ.tE 1tro<; -ro " av " avfJKEt O"'tllV p t1 O"'tllV q avnmoixroc;, evro 1tpo<pavroc; -ro " av " avfJKEt crE oMKA.11p11 'tllV tcroouva11ia (p<::>q). 'E-rm A.ot1t6v exouJ.tE J.Lta cruVE1tayroyiJ a1t6 'tllV o1toia A.Ei1tEt -ro O"UJ.L1tepacrjla : Av (p<::>q), -r6-rE (;)

x2>4 » x>2 x2> 4» x2>4»

x>2 => x2>4). (x >2 x2 > 4). V'xe(2,+oo) ' x2 > 4 », V'xeiR, x > 2 x2 > » IR xe{2,+oo) x2>4» x>2=>x2>4» )

(2,+oo) x>2

3) !:J:ETlKcl J1E TTJV 01t00El�TJ TTJ'ij <Jl>VE1tayroyft<; (p=>q)

D-rav eeAoUJ.I.E va a1tOOE�OUJ.I.E 't11 O"UVE1tayroyiJ (p=>q), ea yp6:q>OUJ.I.E p=>p i => Pz=> P3 =>q ( 1 ) Kat 6xt: . p<::>p i<::> pz<::> p3<::>q t1 aKOJ.l11 XEtp6-rEpa :p<::>p i => pz<::> p3 =>q ME -ro O"UJ.l�OA.tcrjl6 ( 1) yivE-rat aa<pec; (1tp6:y11a a1tapai't11 'tO crE jlta a1tOOEtKnKft otaotKacria), 1tOta Eivat 11 u1t6eE011 p (aq>E'tll pia) Kat 1toto Eivat -ro O"UJ.L1tepacrJ.La q (1tepac;), K6:n -ro o1toio oev yivE-rat Ka-ravo11-r6 O"'tOU<; O"UJ.l�OAtO"J.LOU<; Kat 01tOU U1tapxouv 1taAtVOpOJ.LTJO"Etc;. na 1tap6:oEtYJ.la , av 11ac; i;11'tll eEi va a1tOOE�OUJ.I.E on yta KaeE tcrx6Et 11 cruvE1tayroyiJ :

(2)

(3)

(2)

jaj =5<::>

5+2 1 �

5 + 2jaj 3

=5 <::>

(3),

I � =5

5 + 2jaj 3

Kat

yp6:\jiOUJ.lE

roe;

aeiR

a1tOOE�11

'tllV

1tapaKU't(l)

otaOtKacria:

=5 <::>5 +2 j aj = 1 5 <::>2 j aj = 1 0<::> j � =5 , OEV VOJ.li/;ro 6n ea yivEt Ka'taV011't0

a1t6 1tou �EKtvTJcraJlE Kat 1tou Ka-raA.iJ�aJ.I.E, (<pavllo� ICVKA.o�). EYKAEIAIIl: B' 89

T.l/33

------ rtu tvuv KO\VO T pono tK(j)pUGfl� «nU Mu&qpU't\KU

--------

To aroato A.otnov sivat va ypa\tfouJ.lS: 3a=5+2 l al => 1 3 � = ., 5 + 21 � 1 => 3 l al =5+2 1 � => l al =5. To KaKO sivat Ott OUIJ.�OAlOIJ.O� (2) S1ti x.povta S1ttKpatouas ata CJXOAtKa �t�A.ia Kat OUOKOAa ea a1to�A.11esi. ea x.pstaatsi yt' auto S1tl!J.OvlJ. 0

4) I;XET\KU JIE uopt<JTE� - TUUTOT\KE� E�l<JW<JE�

IIoUoi IJ.a9l')t£� aurx£ouv tl')V aopurrq S�l<J(I)Ol') ' 011Aa0i) tl')V S�l<J(I)Ol') J.lS cl1tt:lp� p�� (1.:6<rt:�) J.lS tl')V TU'UTOTlJOl S�lOffial'), 011Aaoi) tl')V S�lOffial') 1t0U S1taA11erostat Y\« Ka9t: Tlplj TOU (f1)V0Ao1> Opl<rpoo Tl)� (oxt Kat' avayl(l')v yta Kaes x 1R ). Eu9UVTJ yt' auto �£�ata £xouJ.lS Kat SJ.lS�, mpou noW� <pop£� AEJ.lS Ott 11 S�l<J(I)Ol') n.x. 0 . X = 0 sivat aoptatl') 1i tautottKT), oivovta� tl')V SvtU1tffial') Ott ouo MOts� sivat taUtOOl')IJ.S�. To <J(I)crtO ea i)tav va 1tOU!J.S Ott sivat aoptOtl') KUl tautottKT). Kaes <popa 1tOU s11tOUOa ano tOU� IJ.a9l')t£� !J.OU va !J.OU �pouv S�l<J(I)Ol') aoptOtl') 1tSptaaotspot av£<pspav !J.OVO tl')V 0 · X = 0, Av S1tEJ.lSVa va !J.OU �pouv aopl<JTI) S�lOffial') 1tOU va J&t)V t:iVUl TU'UTO'TlJOl !J.S ouaKoA.ia i) uno�o11eou!J.SVot £�ptOKav n.x. � l x l =x , 111lX = 1, x+2r3, x+2'1f+3ro =6 K.A.1t flpO(j)UVcO�: l xl = X � X 2 0 11 !l x=1 � X = K1t, KEZ (Ms tl')V MOta U1taPXSt KEZ, X=K1t) x+2'1f=3�(x,'lf)=(3 -2A., A.), A.eiR Aopurda nproTI)� Ta;tt� i) anJ.Tj aopt<rT{a i) aopl<rT{a pt: tvav t:l.£69t:po cl"(VCO<r'TO, Otl')V 1tpOKSt!l£VTJ 1tSpt1ttffial') tO 'If · ea IJ.1tOpOUOaJ.lS <pUatKa Va ypa'lfOUJ.lS x+2'1f = 3 � (X,'If) = (!J., 3 - !l ), !J.E lR J.lS sA.roespo 6.yvrocrto, to x . 2 X +2'1f+3ro = 6 � (x ,'lf,ID) = (6-2A-31J., A,!J.), A,!J.EIR Aopl<rTia oriTt:pt)� 'tcl;t)� , ourJ.Tj aopl<rTia , aopl<rTia pt: o-6o t:A.ri9t:po1>� ayvro<rTo1>� "' ' ro K.A..n. E1tial')� J.lS axsnKT) OUOKOAia roptOKav S�l<J(I)Ol') aopl<rTI) K«l TUUTOT1.JOl 1tEpav tl')� 0 . X = 0, n.x. tt� (x+ 1)2 (3), .J1 - x 2 = �� (4), J;:i = l xl (1), logx2= 2 logx (2), x2+2x+ 1 llJ..IX + cruvx + 1 � (5) nou tCJXI)ouv yta Ka9s x ato avtiatotxo a'6voA.o opta!J.OU: A 1 = IR , A2 cruvx + 1 llJ..IX llJ..IX = (O,+oo) , A3 = IR .�= [- 1,1], A5 1R - { I Z} . Auto nou tou� ouaKoA.sus totaitspa i)tav to va �pouv s�iaroal') nou va sivat T«1>TOT1.JOl xcop� va t:iva1. aopl<JTI). AVaYKasO!J.OUV tots va tOU� S1ttal')!J.aVro Ott to aUVOAo optO!J.OU tl')� o<psiAst va sivat nsnspaa!J.Evo, o11A.aoi) va !l11V £xst anstpa atmxsia. �sv i)tav roKoA.o Kat naA.t va �pouv t£tots� s�tmoast� 1t.X tl')V � = X , 1tOU £xst aUVOAO OptO!J.OU tO A 1 = { 0} i) tl')V S�lOffial') �- (x 2 - 1 y = X- 1, 1tOU £xst aUVOAO optO!J.OU tO A2 = { X3 E IR/ x2- 1 =0 }3 = { -1, 1 } i) tl')V S�l<Jffial') .jX 3 - X = .jX - X 3 , 1tOU EXSt aUVOAo 3 optO!J.OU to A3 = { xeiR/ x -x 2 0 Kat x-x 2 0 } ={ xeiR/ x -x = 0 } = {-1,0,1}. T£tots� s�taroast� A.otnov anomistat va sivat aoptan;�. E� autrov 11 orotsp11 osv sivat tautonKT), a<pou osv <IA11esust yta Kaes xeA2 napa !J.OVo yta x=1, svro 11 nprofl'J Kat 11 tpitl') sivat tautottK£�. a<pou aA.11eroouv yta Kaes x <no A 1 Kat yta Kaes x ato A3 avnatoixro�. D !lro� osv sivat aoptats�. onro� nposina!J.s, a<pou ta A 1 Kat A3 sivat nsnt::paa!J.Eva. A� napatl')pi)aOUJ.lS teA.o� Ott yta t� avt<rcO<rt:�-aV\<JOt:;l<rcO<rt:� osv £xst V0111J.a xapaKtl')pta!J.O� a6pun17 aviamaf! , a<pou Kata Kavova ot avtaroast� exouv anstps� A.uast�, n.x. 2x- 6 > 0 � xe(3,+oo), x2-4 :::;; 0 � X E [ �2, 2], K.h.1t. !:uyKtKpt!J.EvS� !J.OVO e'xouv 1tS1ttpaa�o aUVOAo AUOS(I)V ' 01t(l)� ot: ii) (x2 -4)2 :::;; 0 � XE { -2,2} , i) .J- X 2 > X - 1 � XE {0} ::s; x 2 + x + 1 � x2+x = O � xe {-1,0} iv) l+ lnx2 2: x2� xe {-1,1} iii) H £wota �£�ata uc� aviamaf! £xst V0111J.a, n.x. : i) x2+1 > x � xeiR = A 1 ii) x+1 2 2 Fx � xe[O,+oo) = A , 2 iii) �- x 2 (x - 1 )2 > x - 2 � xe {0,1} = A3 , 3 J..IX iv) 1l ::; J3 x 1R A4 , onou AI. A , A3 , � ta avticrtotxa a'6voA.a opta!J.OU. 2 cruvx + 2 e

•

10t

K e

•

e x 2 +X

ravror

<::::>

EYKAEIAm B' 89 T.l/34

A ' AYKEIOY

n tlpciJ,IaTa Tuxn� - n 19av6Tt1T£�

AV p�OUf.J£ f.va (,;apt ea E:XOUJ.l.E (l)� f.vost;ll f.vav an6 'tOU� apteJ!o6� 1, 2, 3, 4 , 5 , 6 . 11sv J.11tO pouf.J£ OJ!ID� va yvropit,;ouJ!s EK 'trov npo'tsprov not6v UKpt�OO� apteJ.16 ea od;st 'tO {,;apt. fEVtKa f.va 1tEt pUJ!U <HO onoio osv Jl1tOpoUJ.l.E va npo�AE\j/OUf.J£ 'tO ano'tsAsaJla - Kat a� snavaA.aJl�avs'tat Ka'tro uno n� tOtE� auvei}KE� - OVOJ!a(,;E'tat 1tttpUJ!U TU:XlJ� (1t.T.). Ta nstpaJ!U'ta a'ta onoia 'tO ano'tsAsaJla ;ap'ta'tat U1t0 't� auvei}KE� cm� 01t0ts� yiVOV'tat OVOJ!a{,;OV'tflt atTtOKpaTtKU. :ETTI esropia nteavo'ti) 't(l)V f.!SAE'taf.J£ J!OVO n� 1tteav6Tll'tE� as 1tEtpaJ!U'tU WX'l�· :E'ta napaKa'tro napa8siyJ!a'ta 'tU 1t.T. ava <pspoV'tat as uJoxi9ava sv8s:x6J!EVa I::xoA.w: Ot sK<ppaast� «f.va (,;apt it f.va v6J!taJ!a sivat Uf.J£p6A.tJ1t'tO)) Kaeoo� Kat 11 EK<pPUC111 «sntAt youf.J£ roxaia f.va a'totxsio an6 a'6voA.o OJ!oirow, 011AOOVOUV on 'tU U1tAU EVOEXOJ!EVU sivat taoxieava. g.;

}\ <JKfJ <JfJ 1 .

'EMro n o ottyJiaTtKo� xropo� tvo� xttpaJla•o� 'ri:XlJ� Kat A, B uxorivoA.a Tou n. AivtTat oTt P(A) = 0,5 , P(B') = 0, 3 , P(A' 11 B) = 0,5 . Na 'U1tOAoyi<J£T£ T�

P(A u B)

Kat

P (B ),P( A 11 B ),P( A 11 B' ) ,

P(A u B)' .

AuatJ • •

•

P(B) = 1 -P(B') = 1 - 0,3 = 0,7 P{A' 11 B) = 0,5 � P(B) - P(A n B) = 0,5 � P(A n B) = 0, 7 - 0,5 = 0, 2 P(A n B') = P(A) - P(A n B) = 0,5 - 0, 2 = 0, 3 P(A u B) = P(A) + P(B) - P(A nB) 0,5 + 0,7 - 0,2 = 1 P (A u B)' = 1 - P (A B) = 1 - 1 = 0

A<JKfJ<JfJ

Mapia I:iaKo'U - llavaytroT'I� XptM01tO't>Ao� AuatJ A. i)

X = P(A n B') - P(A) · P(B') = P( A n B') - P( A )P(B') = P{A) - P{A n B) -P(A) P{A) · P(B) = P( A) - P(A n B) - P ( A) [ 1 - P { B)] = P( A )P(B) - P( A n B) = [ 1 - P{A') ] P ( B) - P(A n B) ii) X = [ 1 - P ( A') } P ( B) - P ( A n B) = P(B) - P( A')P(B) - P( A n B) (I) } = P(B nA') - P(A') · P(B) = 3 B. An6 'tO EPOOTTIJ.l.U (i) E:xouf.J£: X = [ 1 - P { A')} P ( B) - P(A n B) .!_ = (1 -.!.) · P(B) _ .!_ � P(B) = 2 9 4 +

l\<JKfJ <JfJ

3. Pi:XVO't>fl£ tva aJltpoA.tJmo �apt ouo <pop� ota ooxtKa. Na IJpdT£ TO 0£tyJ1aTtKO :XcOpO TOO 1t£t paJ1aTO�. 'EMro A: To £VO£XOf1£VO TO a9potOJ1a Trov tvod�trov va dvat 1t£ptno� apt9J16� Kat B: To £VO£XOJ1£VO TO a9potOJ1a T(J)V £VO£i�£(J)V va dvat xollaxMow TO'U 4. Na 'U1tol.oyion£ Tt� xt9av6T'IT�:

P(A) , P (B) , P(A 11 B) , P(A u B) , P(A' 11 B) , P[(A' 11 B) u (A 11 B') ] .

AuatJ

An6 'tov �apaKa'tro nivaKa 8tnA.i}� sta6oou yta T11 1 '1 Kat 2'1 pi'I'll •ou t,;aptou, �picn<ouf.J£ 'tov 8styJ!a-

'E<JTro !l o 0£t'YJ1aTtKO� :xropo� £VO� 1t£tp0.J1aTO� 'ri:X'I� Kat A, B 'U1tOrivoA.a TO'U n A. Av 3 P(A' 11 B) - 3 P(A') · P(B) = l (1), Kat X = P(A 11 B') - P(A) · P(B')

Na axood�n£ oTt:

X = [ 1 - P(A' )] · P(B) - P(A 11 B ) ii) Na 'U1toA.oyio£T£ T'l Ttfl'll T'l� xapaMaOlJ� X. B. Av P( X) =! Kat P(A11B) =! va 'U1tOA.oyi<J£T£

4 4 T'IV 1tt9aVOT'ITa TO'U £VO£XOpivO'U

fta 'ta EVOf:XOJ.l.EVU A Kat B E:xouf.J£: A= { (1 ,2),(2, 1 ),(1 ,4),(4, 1 ),(1 ,6),(6, 1 ), (2,3 ),(3,2),(2, 5 ),(5 ,2),(3 ,4),(4 ,3),(3,6),(6,3 ), (4 , 5 ),(5 ,4),(5 ,6),(6, 5 ) } f.J£ N(A) 18 . B= {( 1 ,3),(2,2),(2,6),(3, 1 ),(3 ,5),(4,4),(5 ,3),(6,2),(6,6)}

EYKAEIAHl: B ' 89 T.l/35

------

f..l£

Mu&qfJ.unKa 'Ylu 'n)V A' A\)KdO'U

N (B) = 9 . Ta svoexoJ.tZVa A,B eivat a<ruJl�i�acrta. l:UJ.lcprova f..l£ tOV KA.amm::6 optcrJlO 'tll� meaVO'tllta� exouf..t£ : N(A) = _!! = ..!_ P(B) = N(A) = _.2._ = ..!_ ' P(A) = N( Q) 36 2 N(Q) 36 4 p (A n B) = o => p (A u B) = p (A) p (B) = 21 + 41 = 43 · Eni<Til�: P( A' n B) = P(B) = -1 4 P [ (A' n B) u (A n B')] = P(A u B) = �4 +

1\ CJKl') CJl') 4. �ntv A' Ta�lJ t:Vo� A-1>Kdo1>, Ka9t: ncnlii pa9aivt:t AyyA-tKa 1\ ralltKa. EntAtyov-r� nxaia tva nat6i 1J nt9avO'n)T« va dvat KopiTCJ\ 1r01> pa9ai Vt:\ AyyA-tKa t:ivat 25% , t:Vcb 11 nt9avO'n)T« . va

«yupl t:\VCl\ , � , t:\VCl\

-35

•

D01.a dvat 1J n•9avO'n)T« va dvat ayopt 1) va paeaivt:\ AyyA-tKa; p) AV CJ1>VOA\Ka 1) A, Ta�l) txt:\ nt:plCJCJMt:p« ano KCl\ A\"(Mt:p« ano 90 naw\6:, 1rOCJ« «Kptpcb� t:\VQ\ T« 1rCl\6\a; Ailcrl')

7�

'Ecrtro x to roxaia e1tlAe"f6J.tZVo na\Oi. 'Ecrtro A: To svoex6J.1ZVO X va eivat a"(Opl. Tote A' eivat tO SVOeXOJ.IZVO X Va eivat KOptt<Jl, Kat B: To svoexOJ.tZVO X va Jlaeaivet AyyA.tKCL. T6te B' eivat tO svoexoJ.tZVO X Jlaeaivet faUtKCt. 25 = ..!_ Kat P( A) = � Llivetat: P( A' n B) = 100 5 4 a) Zl')teitat l') 1tt9avo'tllta: P (A u B) P(A' n B) = � => P(B) - P(A n B) = � . EnoJ.l& vro�: P(A u B) = = P(A) + P ( B) - P( A n B) = �5 + ..!_4 = .!2_ 20 . N(A u B) p) Ewat , P ( A u B) = N(n) => N(A u B) = P(A u B) · N(!l) = .!.2_ 20 · N(!l)Acpou o apt9Jlo� N (A u B) eivat cpumKo� apt9Jlo� npenet Kat o apt9Jlo� �� · N ( n) va eivat cpumKo�. Ene\OfJ ot apt9Jloi 17 Kat 20 eivat nprotot f..1£t�u tou� , auto <ruJl�aivet JlOVo av to N ( n) eivat noAA.anA.amo tOU 20. AllCL Jl� oiOetat Ott: 70 < N ( n) < 90 onote N(!l) = 80 . 0

-------

fta t11 A:U<Til autfJ 9eropt19l')Ke yvrocrt6 6tt av p, y E z Kat a Otatpei to "(\VOJ.IZVO p . y ' a')..).jJ, M.K..::l(a,�)= 1 ' tote a Olatpei to 'Y 'Eva� Jla9l')tfJ� 'tll� A' AuKeiou "(ta va A.i><Til 'tllV Ct<JKl')<Til autfJ xropi� va yvrop�et 'tllV napanavro np6ta<Til npenet f..l£ OoKlJ.l&� va otamcrtrocret 6tt an6 � aKepate� ttJ.l&� N(!l)=xe(70,90) J.16vov '11 x=80 Ka9tcrta to KAacrJla 17x 20 aKepaw. fw Al'yoteper OoKlJ.l&� Jl1topouJle va 9ecrouf..1£ : N(AUB)=y, onote 20y < 90 => 1 19 < y < 153 => 70 N(!l)= 20y < => 17 17 2 2 => 59,5 < y < 76,5 => y E {60,61, ... , 76} => y = 68 (H JlOVl') ano tl� ttJ.l&� aut&� 1tOU Ka9tcrta tO KM20y aKepato ' ). crJla U •

•

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1\CJKl)CJ'l 5 . 'Emro OT\ 1J m9avontT« va ppt�t:' aupto dvat Tp1.1rAaCJl« «1rO TO V« pl) ppt�t:\ K«l 1) m9avO'n) T« V« 91>p1)9cO V« 1rapro pa�i p01> Op1rpEAa t:iVCl\ limMcrL« «1rO TO V« 'n)V �t:'X;aCJID. EniCJl)� «V 1) 1rl9avO'n)T« V« ppt�t:l 1} V« 91>p1)9cO 'n)V Op1rpEAa dva1 90% , nota 1J nt9avontT« va PP«"'ID at>pw '· A

M�

'Ecrtro A: To svoexoJ.tZVO va J3pe�et. Tote A' eivat tO SVOeXOJ.IZVO Va Jll') �pe�el. Kat B: To svoexoJ.tZVO va 9'UJ.11J9ro va napro 'tllV opnpeA.a.. Tote B' eivat to svoex6J.tZVo va �exacrro Va 1tCtpro 'tllV OJ.11tpeAa. Ano un6ee<Til exouf..t£ : P ( A) = 3P ( A') => P(A) = 3 [ 1 - P(A)] => 4P(A) = 3 => P(A) = �4 . Eni<Til�: P ( B) = 2P ( B') => P(B) = 2 [ 1 - P(B)] => 3P(B) = 2 => P(B) = 3_3 Eni<Til�.· P (A u B) = 90% = :0 . EnoJ.l&vro�:

P(A) + P(B) - P(A n B) = _2._ 10 => 9 31 3 2 4 + 3 - P(A n B) = 10 => P(A n B) = 60 Zl')teitat l1 meavo'tllta P (A n B') ,Ol')MOtl '11 mea VO'tllta va J3pe�et Kat va exro �exacret 'tllV OJ.11tp&A.a JlOU, rocrte va �paxro. Eivat: P(A n B') = P(A -B) = P(A) - P(A n B) = 31 = -7 = 23' 3% . -43 - 60 30

EYKAEIAID: B ' 89 'T.l/36

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f\ GKTJ GTJ 1 Aiv£TUt 0 Of:t'YJ.lUTtKO� xropo�

50 10 40% P (A - B) = P (A) - P (A n B) - 100 . 100 B) P (A U B) = P (A) + P (B) - P ( An B) = � 1 00 +

n = {1, 2, 3, 4, 5, 6, 7, 8} evo� n:etpaJ.LaTo� 'T6JCTJ� J.lf: toon:iOava an:A.a f:VOf:JCOJ.lf:VU. 0eropoi>J.LE TU f:VOf:JCOJ.lf:VU: A = {1, 2, 3, 5, 8} AnB = {2, 5 } (1) A B = {1, 2, 3, 5, 6, 8} (2) A) Na ppEiT£ TO f:VOEJCOJ.lf:VO B B) Av yta TO EVOf:JCOJ.lf:VO A t<JXV01>V: A - A = {1, 8} (4) B 4 = {5, 2} (3) N a ppEiT£ TO f:VOf:JCOJ.lf:VO A Ai>oTJ A) B

U '

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10 - 90% 100

P ((A - B) U(B - A))= P(A - B) + P(B - A) = = P(A) - P(AnB) + P(B) - P(BnA) = 150°0 - 110°0 + � - � 80%. L\t6n ta EVOE'V6ueva A-B B A 100 100 eivat acruJlPiPacrta. A) fvropi�OUJlE 6tt: N [ (A U B)' ] = 3 Bpi<rKOUJlE TTtV P( (A U B)') =1-P (A U B)= 1 - � 100 = 1 0%. A.paN(a1t6 KA.acrmK6 opt<rJl6 EJCOUJlE: A U B) ' 10010 N(n)3 �N(Q)=30 p (A U B) ' N(n) ==

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IIapUTTtPoUJlE 6n: (1)�2, 5eB evro 1,3,8 � 6�B 1\pa B = {2,5 , 4, 7 } � Km(2)�46eB' , A n B)� 10010 N(A30n B)�N (A n B)=3 ,7�B 4,7eB E) P (An B) N(N(n) f\aKTJGTJ 3 B) E1tt<Jll <; 1tUPUTTtPOUJlE 6tt: ( 3) =>2,5ellmA 7� 'Emro A.B 01>o EVOEJCOJ.lf:VU evo� E�aUou: ( 4) => 1, 8 � � 3 e Ll. oetyJ.LaTtKoi> xmpo1> n f.l£ P(A)=0,6 Kat P (B)=0,5 A.pa 2,3, 5 � 1, 4, 7, 8 � � , o1t6te m9av6v a) Na oef;ET£ 6-rt TU f:VOEJCOJ.lf:VU A, B OEV dvat UO'UJ.LPiPama 6 � . TeA.tKa L\ = {2, 3 , 5 } it L\ = {2, 3 , 5 ,6}

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f\aKTJGTJ 2 , :Et J.ltU £p£1>VU 11:01> EytV£ J.lf:Ta;iJ TCOV J.LU0TJTrov J.Lta� Ta;TJ� toet;e 6Tt : • To 50% -rrov J.LUOTJTcOV Oa n:ae1 otaKon:t� -ro KUAoKUlpl Of: <<VTJOl» • To 50% -rrov J.LUOTJTcOV Oa n:aet otaKon:t� TO KUAoKaipt <r'TO «lJ01>VO» • To 10% Trov J.1U01)TcOV Oa n:ae1 otaKon:t� oe <<VTJOl» KUl Of: «P01>VO>> • 0 BaoiA.TJ�, o llt-rpo� Kat o AttJ.llJTPTJ� oev Oa n:ave n:o1>0eva otaKon:t� Av en:tu;o't>J.lf: tva J.La&q'T1] 'T't>Jeaia. va 1>1t0Ao'Yl0£T£ Tll V 1tt0UVOT11TU A) Na n:aet otaKon:t� J.lOVO oe <<VTJOi» B) Na 1tU£t TO't>A.UJ(tOTOV Of: tva U1t0 'TU oi>o J.lEPTJ OtUK01tt� . r) Na n:ae1 otaKon:t� J.lOVo mo <<VTJoi» 1] J.lOVo mo «lJ01>VO» A) Na lJptiT£ 11:0001>� J.1U9TJTE� £xtt TJ Ta;TJ. E) . Na ppEiT£ n:ooot J.1U9TJTt� Oa n:ave otaKon:t� TO KUAoKaipt KUt O'TO <<VTJOl» Kat <r'TO «lJ01>VO» Ai>aTJ •

eeropOUJlE ta EVOEJC6JlEVa : A: To EVOEJC6JlEVo va 1tUEt tva<; JlU9TJtft<; OlUK01tE<; <rtO <<VTJ<ri» B: To evoex6JlEVO va 1taet tva<; JlU9TJtit<; otaK01tE<; crto «Pouv6» 50 10 50 ' Tote P(A) = 100 P(AnB) = 100 P(B) = 100

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p) Na oei;tT£ 6-rt:

Ai>aTJ

1� :5 P ((A - B) U (B - A)) :5 ��

a) 'Ecrtro 6n ta A,B Eivat acruJlPiPacrta. T6te P(A U B)= P(A) + P (B) = 0,6 + 0,5 = 1 , 1 > 1 cho1to. A.pa ta A, B oev Eivat acruJlPiPaata. :Euv9enKa: 'ExouJlE P(� P( A) +P(B) -P( AuB) = 1, 1 -P(AuB) > O, a<pou: P(AuB) �1<�1. A.pa A n B 0 (L\ta<popettKa 9a ittav P (A n B ) 0 ) p) A<pou A - B, B - A: acruJlPiPacrta, yta va tcrXt}et: 110 :5 P ((A - B) U(B - A)), apKei 0, 1 � P(A) - P(AnB) + P(B) - P(BnA), it 0, 1 � P(A) + P(B) - 2P (AnB), it 0, 1 � 0,6 + 0,5 - 2 · P(AnB), it P (AnB) � .!.2 , to o1toio taXt}et, a<pou AnB�B�P( A n B) :5 P(B)= !2 . E�aUou: P ( (A - B) U(B - A)) :5 1 .!..!. 10 .

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f\aKTJGTJ 4

AV X e [3,5)

a) Na

KUl d(y,0) :S; 2 an:lon:otTJ0£1 11 n:apama(Jt) :

A = .J4x2 - 4x + 1 + .J,...y':"' . 2 ---4-y_+_4

JJ) Na ppeOd 11 J.LEytO'Til Kat 11 eA.UxtO'Til TtJ.11] Tll�· AilaTJ

'ExouJle: A= .../4x 2 - 4x + 1 + .Jy 2 - 4y + 4 = .J(2x - 1)2 + .J( y - 2)2 = l2x- 11 + IY - 21 =2x-1+2-y=2x-y+ 1 . L\t6tt a1t6 EK<pcOVTJ<Jll EXOUJlE:

EYKAEIAHI: B' 89 T.l/37

-----D EXOUJ.!E: A = c�: + �� :

------ Ma9qJUinKa yt.a 't1)V A' At>KEiou

xe [3 ' 5] => 3 � X => 6 � 2x => 5 � 2x - 1=> =:>2x - 1 > 0 => 2x - 1. E�illou: d(y,O) � 2 => IYI � 2 => - 2 � y � 2 => y - 2 � 0 =::> IY - 2 1 = 2 - y. 3 � X � 5 6 � 2X � 1 0 (I) p) X E [3, 5) 0 (y, 2 d ) � => IYI � 2 => - 2 � y � 2 => 2 � - y �- 2 => - 2 � - y � 2 (11). IIpocr9e'tOUJ..lE Ka'ta JJ.EA.tt 't� (I) Kat (11) Kat naipvouf.te: 4 �2x-y � 12, 8ttA.aoi) 5�2x-y+1�13. H t<rOTTt'ta 2x - y + 1 = 13 tcrxf>et f.lOVOV 6'tav x=5, y=-2 evro tt t<r6'tll'ta 2x - y + 1 = 5 tcrxf>Et f.lOVOV 6'tav x=3, y=2 (yta'ti;). Apa max(2x-y+1)=13 Kat min(2x-y+1)=5. =::}

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A <rKTt<rtt 5

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na Ka9e

X E

) :

= ---xz• (X+ 1 )2 = 1 + K-8 = X2 (X+1 )2 = X2 = X2X2 + K-8_ X2 x2 na va eivat 11 napacrta<rTt A ave�ciPTTt'tll Tou x 8 npenet Kat apKei: ';z = 0, 8ttA.aoi) K = 8, a<pou x * O. 'E'tcrt EXOUf.lE D = lR - { 0, - 1 } . X2 + 2x+ 1

(X+ 1 )2

x2 +K-8 x2 {X + 1 )2 (X 2 +K-8)

(X+ 1 ) 2 X2 +K-8

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A <J KTt <JTt 8 AivE'tat tt t<JO'fll'T U:

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A) Na a1ro<iE�E'TE O'Tl : x·y � 0

B) Na a1ro<iE�E'TE o'tt : x Auatt

�1, x,y e R (1)

l x + t IY

A) Enetoi} l x l + 1> 0 Kat IYI + 1 >0 TOTE apt9f.loi x, y Eivat Of.lO<JTtf.lOt i) f.lTtOEv, 8ttA.aoi) x·y ?: 0. B) A<pou X y>O ea EXOUf.lE: (1) ...l!L = J!L l x l + t IYI + 1 => l x i i YI + I x l = l x i i YI + I YI => l x l = I YI => x = Y =>

•

O'Tl: AaKqaq 6

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= ----- + 4a2 +4a+ 1 l 2a+ 1 1

' 01tOtl a * z' - 2 A) Na a1to<iE�E'TE O'tt : A � 2 B) Na � av m 2 dvm � np.1l 'TTt.; A. A6att 4a2 -4a+ 1

l 2a- 1 1

'EX;OUJ..lE .. A

A <J KTt <JTt 9 A) na Ka9E 9E"TtKo apt9p.o x, va a1ro<iE�E'TE

+ - � 2. 1

B) rta 'tou.; 9E'ttKou.; apt9p.ou.; a, p, y va R+y-a y+a-R a+ R-y 3 I I' I I' > U1tO<iE:J:E'TE O'T\" I'

A6att

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2 � O =::> x + -1 � 2, 1 x+-2 = x2 + X1 - 2x = (x -1) X . X f.lE 'tO icrov f.lOVO yta x= 1. B) Av A 'tO nproio f.lEAO� 'tll � 'tO'tE: A= + + + +

(2 a + 1)2 1 ( 2 a-1) 2 l 2 a + 1 1 2 1 l 2 a + 1 1 2 1 2 «+ 1 1 l 2a-1 1 1 2 «+ 1 1 1 2 «- 1 1

H(! ;) (: �Hi �) + -

=1 2a + 1 1 + l 2a - 1 1

E<papf.l6/;0UJ.!E 'tTIV tOtOTTt'ta lxl + lyl � lx + yl a. � 'tO icrov f.lOVOV 6'tav = 1, � .!_(2 2 2 3) = yta x = 2a+ 1 Kat y =-2a+ 1 on6'te exouJ..tE : + + 2 p A= l2a + 1 1 + l -2a + 1 1 � l (2a + 1) + 2 (-2a + 1) 1 = 2 . Apa A � 2. � = 1 , 1... = 1 , ottA.aoi) a=p=y. y a. B) To io"ov 1CJXI)a J.U)vov 6mv(2a + 1 ) ( -2a + 1) � 0 , A <J KTt <J Tt 1 0 1 1 a. -2,.2 . }\pa 'tO 2 etvat EJI.UXt<rTTt 'tlf.lll a) Bpei'TE 'ta (1 - -J2 )3 Kat (1 + -J2 )3 'tll� A Kat A.af.lPave'tat yta � anetpe� Tt!Ji� 'tOU p) Na <iE�E'TE 6n : 3_J s-J2 + 7 - 3_Js-J2 - 7 = 2 Auatt a. e n.x. yta a. e 0, � , - ± . a) 'EXOUf.lE: (1 - ..fi. )3=13 -3· f.J2 + 3·I.J22 -.J23 = A <J KTt <JTt 7 = 1 - 3J2 + 6 - 2J2 = 7 - 5J2 2 1 (X+ 1 )2 'EmiD: A =( 1 + - �): -- . Of.lOtoo� (1 + ..fi. )3 = 7 + 5J2 x X2 x2 +K-8 A) Na ppE9£i 'TO cr6vol..o optap.ou 'Til.; p) Vs..fi. + 7 = � =I+.J2 , evro 5J2 - 7 = B) Na ppE9£i 'TO Ke R romE tt 1rapumamt A va dvat avE;up'fll'fll 'TOt> x. = ( J2-1 r =>�5J2-7 =.fi - t : Apa Auaq A) :EuvoA.o opu:rf.tou 'tll � A eivat 'tO Vs..fi. + 1 - Vs..fi. - 7 =I+.J2 - ( .J2 -t) =2 D = { x e lR / x * O, x 2 + K - 8 ;t: O, x + l ;e O } . M-.'l n�l v• r""'-''

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H O'IYyKptOTI tptyrovrov a1totsA.ei �acnK6 Kat t<JXl)pOtato spyaA.eio SmAUOTI� 'YSOOJ.LStptKIDV 1tpO �A1)J..UltOOV. l:s J.WptKe� a1t6 tt� Mast� trov aO'ldjas rov 1t01) aKOAoU90UV, 9a 9sropf1aOUJ.LS yvroat6 Ott tO a9potOJ.La tOOV yrovtiDV KU9S tptyiDVOU StVat 180° .1tpUyJ.La to 01toio a391tOOetKV'6etat ro� yvro ot6 J.LStU f11 9sropia tOOV 1tapaA.A.f1A.rov eu9etrov, Ka9<.0� Kat f11V BwOta tOU 1tapClU1)AoypUJ.LJ.LOU.

Kat :I:f'Z = f'1 + f' = 30° + 75 ° = 105° . 2 AO"Kl)Gl) 2 l:E -rpiycovo ABr, (j)tp01>JI.E 'TO U'JIO� AA. l:'f11 paaq Br 1taipvol>pE aqp.do E, ro«J'TE: AE=AB. I:.:xEota�co

-r�

ycovu�

rAll = 2ili . AE\;'TE o-rt

AEH = AEA Ai'H = :8 •

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Kat H

l:'f11v 1tpotK-raaq '"1� 1tA.E1>pa� AB -rnpayro vo'U ABrA, 8Ecopo'6pE aqpEio E, ro«J'TE BE=AB. ME paaq '"1 BE axEota�o1>pE G'To ioto 11JI.tE1ti1tE oo pE -ro -rnpaycovo, -ro tao1tU'Upo -rpiycovo BEZ. Ta 'Tp'l\pa-ra AZ,BA -rtp.vovtat G'TO aqp.do I:.. AE�'TE 6-n -ra -rpiycova I:.AB,I:.rz dvat iaa. r

Auo-1)

t:.

Eivat cpavsp6 6tt AE=AB Kat & = B , acpou J.LEOOKa9sto� tou BE. To tOOOKEAB� tpiyrovo ABE exst BAE = 2cp = rAH . fta va sivat Aol1tOV Kat AtH = .B , apKEi Af=AH. IlpayJ.Latt ta tpiyro va ABf' AEH sivat iaa, Ka9ro� exouv: A'6 01) 1) AB=AE, A1t6 to taoaKsA.e� tptyrovo ABZ (AB=BZ) Kat 2) B = AEH Kat BAr= EAR = EAr + 24> . Apa S1tSt0f1 ABZ = 90° 30° = 120° 1taipvOUJ.LS AH=Af. A I = zl = 30° . Ta tpiyrova ABl:,Bfl: sivat iaa, AGK1)CJ11 3 To -rpiycovo ABr dvm. op8oyrovto G'TO A Kat ot6tt exouv: t«JOGKEA.f�. 1 ) AB=Bf, (1tl..eupe� tstpayrovou) 2) Bl: KOtvf1 Kat 3) B1 = B = 45 . l:uve1tro� 2l:A=l:f Kat f'1 = A1 = 30° . Ta tpi yrova Bl:Z Kat BfZ, sivat S1tl011� iaa Ot6tt exouv: 1) sz Kotvf1, 2) zl = .s3 = 30° Kat 00 = rzB = 7 5° (to tSAeutaio 1tpOKU1ttSt a1t6 tO taoOKsM� tpiyrovo BfZ ). A1t6 f11 tOOf11ta o.utrov trov tptyrovrov 1taipvouJ.W: fZ=l:B Kat l:Z=Bf(=AB). TsA.tKa ta tpiyrova Al:B Kat l:fZ exouv Kat � tps� 1tAeupe� tOU� ias�. Apa sivat A1to tva nxaio aqp.do I:. '"1� 1tA.E1>pa� AB, iaa. (j)tpco Ka8E'f11 1tpo� '"1 Br, 11 o1toia -rtp.vEt 'f11V M1topo'6J.W va a1tocp'6youJ.W f11 O'IYyKptOTI trov 1tpotK-raaq '"1� rA G'TO aqp.do T. AE�'TE O'Tt tptyrovrov Bl:Z, BfZ, acpou ta tpiyrova Al:B, Bi'l: = Bf'l: . l:fZ exouv l:A=l:f, AI = tl = 30° Auo-11 To tpiyrovo Al:T eivat Kat auto op9oyrovto Ai:B = 180° - AI - .8 1 = 1 80° - 30° - 45° = 105° Kat lOOOKSAe� (1tpOKU1ttEl wt6 tl� Kata Kopucpf1v M

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yrovif:<; l:). E6KoA.a rtJ...iov otartu:m:Ovro, on 'ta op9oyrovta 'tpiyrova AT� Kat ABT eivat icra (Ar=AB, �=AT), Erto�ro<; m = ITA Kat 'tO'tE: ({J =, a<pou eivat � = 45° m ro = 45° -=ITA B 'T p orto�: llpo<pavro<; 'to l: dvat crTIJlEio 'tOJlTJ<; ouo U'IfOOV 'tou 'tptyrovou TBr. Apa l:r ..l TB , orto'te ro=<p ro<; o�eif:<; JlE nf...eupe<; Ka9e'te<;;. -

AOKfiOfl 4

:tTO &O(l)T&ptKO TO\) TptyroVO\) ABr, JlpioK&Tat OT)JldO :t, rooT& :tA=6, I:B=4, I:r=s. KaTaOK&\>U�Ol>Jl& 7tp0� TO ptpo; TTJ<; AB Ta t007tf...&l> p a Tpiyrova ABA Kat B:tE. Y7toA.oyiOT& To p1]Ko<; TO\> TJ11]paTo<; AE. A

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rupo6 f:xmN f:1t11rJ...iov M I = M 2 m<; Ka'ta KOpuqn1v. Apa �'=ME', rtpoKfurrm & Kat �=ME. B 'T p orto�: llpo<pavro<; ��'/lEE' Kat a<pou oe�aJ.I£ on ��'=EE' 'to 4-�'EE' dvat rtapaUTJA.OypaJlJlO, o rto'te ot �E.�'E' otx,o'tOJlOUV'tat cr'to M. f 'T p ono�: Av J.l£ M oev cruJl�oA.icrouJ.I£ 'to crTIJ.I.Eio 'tOJlTJ<; 't(!)V �E. �'E' aUa 'tO JlE<JO 'tOU �'E', 'tO'tE apKEi va oei�OUJ.I£ O'tt 'ta �.M,E dvat cruveu9etaKa, OTJ A.aoiJ �ME = 1 80° . llpayJlan 'tO'tE Kat rtaA.t 'ta op9oyrovta 'tpiyrova �M�' Kat EME' dvat icra, a<pou, M' = EE', M�' = ME' . Apa M 1 = M 2 , ortO'tE �E = M3 M 2 = M 3 + M 1 = 180° . +

AOKfiOfl 6

llavro OTO T&TapToKVKA.to

OAB , Jlp(OK&Tat

OT)Jl&iO r, rooT& AOf = 20° . E7ti TTJ<; or 7ta(p V0l>J1& <tl)p&io T, roOT& rT=rA. rpa<pOl>Jl& TO T& TapTOKVKAlO (O, OT) . H p&ooKaOno; TO\) or, TEJIV&l Ta �VO T&TapTOKVKAla KUTU O&lpa OTa <ti)Jlda P,A. A&i�T& Ot't Ta Tpiyrova OPA Kat OTA &iva1 (oa. A'6 01J

EmJ...iyro va cruyKpivro 'ta 'tpiyrova �EB Kat Al:B, cr'ta onoia eivat npo<pavro<;; B�=BA Kat BE=Bl:. ArtOJlEvEt va od�ro on EM = fiiA 'tO onoio eivat aA.TJ9E<; a<pou Kat ot ouo yrovie<;; tcrouV'tat JlE: 60o - EBA . TeA.tKa �E=�=6. '

AOKfiOT) 5

I:To toooK&Ai:; Tpiyrovo ABr(AB=An, 9& ropoup& OTJJldo A TTJ<; 7tf...&l> pa; AB Kat OTJJldo E OTTJV 7tpotKT«OTJ TTJ<; Ar, roOT& BA=rE. Ad�T& Ot't TO AE �hx,oTop&i TTJV 7tpofloJ..1] TO\> A 'E ' &7t( TTJ<; Br. A'6 01J

Ai>oTJ H ap a'T1]pTJ 01J : na r17 ).:6071 r17<; 6.o7a!071<; atro7<; Oa XPYJUZfl07rozfJaovpe Kaz r17v aprpifJo).IJ 7tepbnwU1f uTo nrraq Tpzrwvwv aVfl({JOJva pe r1JV o1roia av

'Ecr'tro �' ,E' ot npo�oJ...i<; 'trov �.E cr'tTJ Br Kat M 'tO crTIJlElO 'tOJlTJ<; 't(!)V �E. � 'E'. (� = 13', y = y 1, t = t�) rare ra rpiywva eivaz iaa fJ B B 1 = 1 80° . llpo{{JavdJ<; av oz ywvie<; B, B 1 eivaz A

Kaz oz ooo oi;eie<; fJ Kaz oz ooo apfikie<; a1rol<kiemz va eivaz B + B 1

= 1 80° , o1rme rpiywva eivaz iaa. Ot yrovif:<; 'tou tcrOOKEA.o'6<; 'tptyrovou ITA eivat 80° , 50° , 50° , cruvertffi<; fAO = 80° - 50° = 3 0° . E rticrll<; A.Oyro 'tTJ<; J.�.EcroKa9ttou 'to 'tpiyrovo �or civat tcr6rtA.eupo, ErtoJ.!Evro<; PW = 3 0° . Ta 'tpiyrova, A.ot rtov, OP� Kat OTA, f:xouv O�=OA (mcrive<; 'tOU iOt TO'te 'ta op9oyrovta 'tpiyrova M'B Kat EE'f civat ou Kt'>KA.ou ) Kat OP= OT ( yw 'tOV iOto A.Oyo). Erticrll<; icra aq>OU, �=Er Kat B = t = f1 • A.pa M'=EE', o ot yrovif:<; .fi)O , ArO dvat aJl�A.Eie<;. l:t'>J..L<prova J.1£ 'tTJV rtPOTJYOOJ..lEVTi rtapa'TiJPTlcrTI 'ta 'tpiyrova rtO'te 'ta op9oyffivta 'tpiyrova �· Kat EME' civat icra A.omov eivat icra. E

EYKAEIA� B ' 89 T.l/40

AaKtiatl� )\Aytf}pa� - I:uvapTtiatl�

B ' AYKEIOY

�KOt'iOa<;; �ron]pto<;;

ACJK11CJ11 1 : (�)f:rop01)J1E 11] 'YV11tJlO><;j J10VOTOV11 <n>vapT11<fiJ

f : R -t R JIE f(R) =(-1,1). Na

anoo£tX9£i

f g(x) = 1 (x) 2( + f x)

OTt

<n>vapT11<fiJ

dvat 'YV'Ilairoc;; JIOVOTovq JIE

tOtO ElOO<;j JIOVOTOVia<;; JIE a'UTO 11]<;; f .

B) Anooti;Tt oTt uJxvtt

R. Eivat o T11<;; g(x) ;

apt9J10<;i

g(x) � 7i'

7i'

yta Ka9t

11 flkyt(J1'1] TtJ11]

Ava11 : A) Av 11 f sivat "fVll crlroc; cpeivo'Ucra Kat x r , x2 e R � x 1 < x2 'tO'tE SXOUJlE:

g(X I )-g(x2 ) . . =

2 )) (1 - f(x1)f('S)) >O' . ( f(x1)-f(x (1+f (x1 )X1+f ('S))

3t6n f(x1 )>f(x2) Kat - 1 <f(xt)< 1 � lf(xt)l < 1 . OJ.!oiroc; lf(x2)l < 1 . Apa lf(xt)l · l f(x2) 1 < 1 � l ftxt }ftx2) 1<1 � 1 <ftxt)·ftx2)<1� 1-ftxt)·ftx2)>0. Apa g(x1 ) g(x2) 3TJA.a3fJ, TJ g sivat "fVll O"troc; cpeivoucra. OJ.!oiroc;

f i JR. � g i JR. 1\

E6KoA.a

•

3ta1ttO''tcOVOUJlE

- IlapapTTIJlU E.M.E. Kap�iToa�

6n 11 f sxst 'touA.axtcr'tov 3uo (3ta<popsnKsc; cpucrtKa) p�sc; cr'to t!, ac; sivat ot P I < P2 . Av fi 'tO'tE PI <p2� f(pi) < f(p2) �0<0, U't01\

..i-d, V

1to. AvaA.oya: Av f 'tO'tE PI <p2� f(p t) > f(p2) �0>0 , U't01tO. Tt O'TJJlaivEt am6 ysro� 'tptKa; Ma cpucrtKa 6n 11 ypacptKfJ 1tapacr'taO'TJ 'tTJc; f 'tSJlVEt 'tO 1tOAU Jlla <popa 'tOV a�ova X ' X O'E O'TJJlelO JlE 'tE'tJlTJJlSYTJ O''tO ti . 211 llapaTftP11CJ11 ! Av yvropi�OUJlE 6n TJ cruvap 'tTJO'TJ f eivat "fVll O'troc; au�oucra O''tO 3tUO''tTJJla ti Kat a, b Et!, 't6n: f(a)<f(b) "*a a>b.

ACJK11CJ11 2 Av g(x) 'YVt)Giroc;; av�o'Uaa <n>vap11]<fl) Gt'O R JIE g(1) = 1 Kat f(x) = g3 (x) + 3g(x) - 3 yta Kcl9£ X E JR a) Na O£i;£T£ 0Tl 11 f dvat 'YV'IlCJiroc;; av�O'UGa. p) Na AVGET£ 11]V aviCJO><fiJ

f ( f ( x )) < 1 .

A1toou�11 : a) Ka9roc; ot 1tEpt't'tsc; 3uvaJletc; Kpa'tavs ne; 3t a'ta�stc;, ea SXOUJlE yta Kaee X I ' E ' x r<X2 3 3 �g(xt)<g(x2)�g (xt)<g (x2), 3g(xt)<3g(x2) � f(xt)+3 <f(x2)+3� f(xt)<f(x2). llapa'tTJpou� 'tcOpa O'tt f( l ) 1 , E1tOJlSvOOc; f ( X ) 1

x 2 JR. 1 g(x) = [rcx>-1t O� g(x) 1 yta Ka9E . � > 2 22 [1+ f2 (x)] , , 'tO 21 JlE"{tO''tTJ , OJlroc; X E � rta Va Etvat 'ttJlTJ, (f ) < � 'tTJc; g(x) apKEi va u1tapxst x e D8 = JR. rocr'tE f { f(x)) < f(l) � f(x) < 1 � f(x) < f( l )� x < 1 311 llapaTfJ p11a11 : Av 11 cruvap'tTJO'TJ f SxEt 'tO ig ( x) = _!..2 . D J.!roc; g ( x) = _!..2 f{ x) = 1, Kat eivat 3to si3oc; JlOVo'toviac; crE 3uo 3tacr'tfJ Jla'ta (a,x0 ), (x0,�) ti t , ti2 AEN <nJJ.laivtt 6n ea MUVa'tTJ acpou f(R) (-1 , 1 ). Apa 'tO _!.. 3ev ei2 sxst 'tO i3to si3oc; JlOVO'toviac; Kat O"'tTJV tvrocrfJ 1lD

•

{:::>

vat J.!Sytcr'tTJ 'ttJlTJ 'tTJc; g(x). Am6 6Jlroc; 3ev a1tocleist 'tO ev3sx6J.!EVO va sxst J.!Sytcr'tTJ 'ttJlTJ

'touc;. fta 1tapa3styJla 11 . cruvap'tTJO'TJ f(x)= l /x sivat "fVll O"iroc; cpetvoucra crs Ka9tva a1t6 'ta 3t-

_!.. H 3ta3tKacria yta va a1to-

acr'tfJJla'ta (-oo, 0) , (0 , +oo) alla O"'tTJV SvOOO'TJ 'touc; t!EN sivat JlOVO'tOVTJ, acpou f( -1) < 0 < f (1) evro - 1 < 1 .

JltKp6'tEPTJ 'tOU

•

cletcr'tEi fJ 6xt Kan 'tS'toto sivat apKE'ta ev3ta cpspoucra (ytari;). 111 llapaTft P11CJ11 : Av TJ cruvap'tTJO'TJ f sivat "fVll O'troc; JlOVO'tOVTJ O'E tva 3tUO''tTJJla ti, 'tO'tE sxst 'tO 1toA.u Jlta pi�a crE au't6 'tO 3tUO''tTJJla . na-ri ; H a1t63e�TJ Jlac; ea EtVat SJl�O'TJ· 'EO''t(l)

EmO"TJc; ac; 1tpocr�ou� 6n av yta Wo cruyKE KptJ.!Svouc; apteJlOUc; a, b ev6c; 3tamfJJln'tDc; t1 � a>b tcrxUEt fta)>ftb) AEN <fi)Jlaivu 6n civat "fV''lcrtroc; au�oucra O"tO ti, EKL6c; av yvrop�OUJ.!E On •

EYKAE� B' 89 T.l/41

------

Ma8fUUlTtKa yw Tqv B ' A-uKtio-u

tiVat yvt'l� f..lOV&tOVfl. 4'� llapaT'I]pt]mJ : Y1tapxouv cruvaptftcrs� 1tou o ev sivat yvTJcriro<; f..lOV6-rovs<; crs Ka.vtva. ota cr'tll f..lU . fta. 1ta.paostyf.1a. 11 cruv«ip'tll crTJ Dirichlet

f(x) = {1'

X Pfl'tO� X appfl'tO� ,

' t1EN swat yvTJcrtro<; f..lO-

v6-rovrt crt Ka.vtva. U1tOOt«icr'tll f.1U mu R, ot6n Ka.ero<; -ro R tivat 1t'UKV6 cr-rou<; P11to'6<; Kat -rou<; app11-rou<;, ea. sxouf.1s on ym x1 < x2 p11-ro6<; ea. SXOUf..lE 6n U1tapxst UPP11't0<; X3 rocr-rt X}<X3 <x2 o1t6-rs f(xi) 1 > 0 f(x3) Kat f(x3) 0 < 1 f(x2) ' apa OEV U1tapxst cr-ra.SspiJ OlU't�11 't(l)V stK6vrov. AGKt)mJ 5 !:To <JXftpa 1>1tciPXEl 1) ypa<ptKft =

1tapamamJ 'Tt)� G1>V U P'T1JmJ� .

1 <p(x) = - · x 2 2

Na <JXEOlQG'TEi 1] ypa<ptKft 1tapamamJ 'Tt)� G1>VU P'T1JmJ�

-------

(<ru f..lf.'t.lt plKTJ 'tll <; g ro<; 1tp0<; 'tOV U�OVU X ' X Kat f..lt'tU Ilia optS6vna. f..lt'tUt61ttcrTJ a.utft<; 1tpo<; -ra.

2 f..lOvaos<;.

E1tof.1£vro<; - g (x) - f(x) 011A.a.oiJ f(x) + g(x 0.

os�ta Ka-ra

g(x AGKt)Gt] 7 0 sropo'6 ps 'T1) G1>VUP'T1JmJ f(x) = - x2 + 4. MET«Ktvot>ps 'T1) 1t«papol.ft y = f x) optSo=

(

v'Tiro� KCll KU9f'Tro� cOG'TE 1) K0p1><pft p Cl1>'TTt� VU f1E'TClKlVElTCll 1tclV(l) G'Tt)V E1>9Eia (E) y = + 4 G'T1) 9tm} Q, 01tol� OEiXVEl TO 011tAClVO <JXftpa. H vta 1tapapoJ.ft Ttpvn 'TOV a�ova x'x ma mJpEia B, C 61to1> A, B T« mJpEia 'TO flit� 'Tt)� y = f pE 'TOV a�ova x. Na

( X)

pps9o'6v 01 G1>V'TETayptvs� T01> C.

[L11aywvu;poq ((Euclid)), Kavadaq - Waterloo, 2000]

1 h(x) = - · x 2 + 2x + 3 2

A'6Gt] 'Exou f.1t: A'6mJ

�(

h (x) = x 2 + 4x + 6} = = � ( x2 +4x +4+2} = �[( x +2)2 +2] = � ( x +2)2 +

Tia.pU'tll POU f..lt 6n

f { O ) = 4 Kat f { x ) = o x = ±2 . Apa. YP = 4, x A = -2, x 8 = 2 . Av esropiJcrouf..lt 6n 11 f

�

f..lS'tUKtvTJe11KE 1tpro-ra. opts6vna. KU'tU c f..lOVU OE<; Kat f..lE'ta Ka.-ra.K6pucpa. Ka.-ra k f..lOVaos<; (c, k>O), 'tOtE 11 vsa. 1tUpa.�oA.iJ ea. SXtt t�icrrocrTJ y = g x f.1S g(x) f(x - c)+k=-(x-ci+4+k

( ) Kat x Q = Xp + c = O + c = c, YQ = yp + k = 4 + k . AUa Qe(s) � YQ = x Q + 4 � 4 + k = c + 4 � elf! 2 o1t6-rs � k = c � g ( x ) = - ( x - c )2 + 4 + c , g ( x8 ) =0� g ( 2) = 0�- { 2-cl +4+c = O � f � -4 + 4c - c2 + 4 + c = O � c2 - 5c = 0 � c = 5 � (fiC'!(f'l 6 !:TO <JXftpa 2 a.cpou c>O . TtA.tKa g ( x ) = - ( x - 5 ) + 9 , o1t6-rs TCl PAt1t01>pE ypa<pftp«T« ouo --��----�x• g( x) = O � x-st +9=0 � x-st =9�x=2 ft x=8. G1>V«p'TftGE(l)V f, Apa. e(8, 0) . g. ll01a Eivat 1)

01t6-rt h ( x) = <p( x + 2) + Apa. 11 ea. 1tpoci \jltt 1tpOO'tU U1t0 f..llU OPlSOV'tlU f..lt'tU't01tlcrTJ 'tll <; KU'tU f..lOVUOE<; 1tp0<; 'tU a.ptcr-rspa Kat f..lE'tU U1t0 f..llU KU'tUKOpU<p11 . f..le'tU't01tlcrTJ 1 f..lOVUOU<; 1tpo<; 'tU 1tavro.

<JXE(J'J} JIETa�\) TroV f , g ; [A1to -ro A•arwvl(fpo <<KarKovpm" 2009] A'6Gt] Tia.pa.'tll pOUf..lt O'tl 11 f ea. 1tpOKU\jiEl U1t0 'tllV - g

•

Na A.uesi -ro iow 1tp6�A11f.1U Kat 6-ra.v (s): y = X + 0 ' rocr-rs vu f..l11 . 011f..llO'UPYTJeti 11 t VTU1tfficrTJ 6n 1tpoK'61ttEt 1tav-ra. k=c.

EYKAEIAIIl: B' 89 T.l/42

------ Ma&qpanKa yt.a ntV B' At>Kdot>

------

B ' AYKEIOY I:KoTi6a� I:roTilpto�- IlapapT1lJ1« E.M.E. Kap6iTaa� 1 . Na J.v9£i TO ri<J't'1')p« 6751 X + 3249 y = =26751 3249 X + 6751 y = 23249 [E. M. E. 1986]

Aua11

�TTIV 1tEptnTOOcrll 1tOU ak + �m 0 , naipVOUJ.I£ a = - � i( Kat 01 E�tcrrocrEt� yivoVTat mx -ky = 1 , mx ky = � 01tOTE av � -ky TO (�) Eivat aouvaTO, OTJAUOll 01 EU9EiE� Eivat 1t(l pciUTJAE�, evro av � = - ky , OTJMOll av cruvol.tKa � = _1 = .r TOTE 01 EU9EiE� TaUTt�OVTat (mx-ky= 1) m k 1 =

JlEAll naipVOUJ.I£ 10000x+10000y=50000=>x+y=5, evro a�atproVTa� naipvowu: 3502 x - 3502y 3502 => x - y = 1 . IIpocr98ToVT� n� vsE� E�tcrrocrEt� naipvot>JlE 2x=6=>x=3=>y=2 (OEKTE� TlpE�) KUTcl

Ilpocr9ETOVTU�

m = k-k = O , m k - , D = ){ m2 k2 = 3 m3 , Ym k2

D= k

k K

m k E1tOJ1EvOO�, av k m ea Eivat D 0,01tOTE TO crU O'TTIJlU EXEl JlOVUOtldJ Mcrrt TTtV (x,y)= ( �· ,� ) = (o , J{m ) · Av k = m , TOTE 01 , TUUTt.,OVTat TTIV + y = 1 , apa , uUO E..,tcrOOcrE� 2k s:: ,

TO (

---

(�)

J.l£

EXEl

x, y) = ( x, :2 -x ) , x e R

3. AivoVTat 01 tl>Ot�

(t1) Kat (t2):

---

1 x--

cl1tEtpE� AUOEl� (k crTa9Epo�)

(tt):ax+py+y=O( I«I + IPI�O)

_____m = .I_ km :;t: O . k

( m

)

{

D = (if7 -1)[{ if?r +if7+1J- (�r = ( if?r -13 -3 = 3

Kat avciJ.oya Dy=-3 Kat D, =2(�n�+.if7+1)=k.

Apa TO (�) sxst Jlta Mcrrt TTtV (x , y) = ( �, -1 )

ppdTt O'l>V-

@ip.a OAl')'olmrrov test a'wJ..O')'IJmJr;: Na ypa\jlete tva ypaJ!JltK6 ml<YtT]JlU (Jle ayvrocrrou� crro rutoio va eivat: 0, 0. 'Exm AUO'e� au-r6 TO ml<YtT]J!U; Av vat, 1t0�; Ka9e J!U9r!n1� Jl1topei va ypa\jlet Ilia ota<po pen!d] aJ.J.iJ. crcoO'ti] anllvrrt<JTJ apKei va 7tapet: y a _ y a . - =1= - ' - =1= - ' ' 61tco� a' W W y' a' y MH rPAMMIKA I:YI:THMAT A 6. BpEiTt ou� T� Tpta6t� (x, y, z) xpaypaTtKrov apt9prov 1r0l> lK«VOXOlOUV T� <JXtGt�

2x2 D:;:z::O, Dy = D, :;:z:: � �

x, y)

+ 2y = 6 {3x6x+5y=12

---

4y 2 4 x2 4z2 z y = ' 4 y 2 + 1 � ' 4z 2 + 1 = x 4 x2 + 1

[OJ..vp.xuida Kavada 1996]

Aua11

91lKt� roO"Tt 01 (tt), (t2) a) Na TtpvoVT«l p) Na Eivat xapall1JU� Na T«VTi�oVTat Aua11

} t'3 · x + { \149 + � + 1 ) y = \149 + � + 1

Aua11

Dy =

{{

4. Na J.v9Ei TO ri<J't'1')p« � - 1 x + ifi7y = �2187 ·

6uJ.(j)opt� Ttpt� Trov k , m E R*

:;t:

Av x=O TOTE y=z=O Av x 0 TOTE y,z 0, 01tOTE 4x2 +2-,1 11, -1 = 1 + (-1 )2 . 0Jl0100� ano ano, (1) -1 =-y 4x y 2x (2), (3). ME npocr9Ecrrt KaTa JlEATJ EXOUJlE: :

+ �y = -y va E- 1- 1 ' 1, ApKEt, npo�avro�, TO crt>OTTtJl« + (1- = 0 ( 2. ) + ( mx ky = 1 XEt a) Ilia �) KaJlia y) anEtpE� J.ucrEt� avTicrTotxa Av D = l am -�k I = - ( a k + � m ) 0 TOTE TEJlVOVTat Apa x = y = z = _!_2 (OEKTE� TtJlE�). 7. Ilmo Eivat To xJ.-qOo� Trov xpaypaTtKrov J.u crE crTIJ.i£tO TOU 01t0tOU 01 crt>VTETUYJlEvE� EtVat 11 atrov Tov O'l>anlJl«To� {lxl + lyl=l, x2+y2=a} (I:) (JlOVUOtldJ) Mcrrt TOU (�) ax

EYKAEIAHI:. B' 89 T.l/43

;J'

------

Ma6qpanKa y1a TI)V B ' A1>K£ioo

"(la � �t.a(j)opt� npayJlaTt.Kt� Tt.Jlt� -rov

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( 1)

1 £X£1 'tO't£ · · { I } t = 1 ± .J2a"=1 r . 1(1 a E 2, u's:U• O 2 Ota<popenKe� p�e� k,m, 01tO't£ Kat 'tO (l:I) exet Oi>o AV �0 'tO oUO'tl)J. (l:) eivat aouva'tO. 'Ecr'tOO ota<popenKe� A.i>cre� (k,m). EnoJ,ttvoo� 'tO apxtK6 A.otn6v a>O. OK'tcO AU0£1� (x,y) . oUO'tl)J.l{l (l:) BX£1 To 0'6cr'tl)J.la avnJ!e Tooni�e·tat aPXtKa ·� l:xoA.w : Av 1t.x. a = % · T6te (�) �(�m) e · � yeroJ.le'tptKa oo� e�i)�: H ( 1) naptcrravet 4 euW � -� � -� 01t0TE: (r) � (x y ) ypaJ.lJ.l(l 'tJ.li)J.la'ta 'ta onoia ppicrKOV'tat enl 'tOOV 4 3 I � _! --.3 I --.3 _! . fta a= l TO (r 1 ) &.et 1tclA.t ouo 9ruOV y=x+l, y=x-1, y= x+l, y=-x-1, <Wt6 ot6otacpopeTuce� Mcret� (k,m) (1,0),(0, 1) . Oflro� TO (L) sxet n O:s;l X I= I- I Y l => (I:x. t) flOVo TEcrcrep� Mcre� (x,y) n� (1, 0),(-1, 0),(0,1),(0,- l ), =>l y l:s; I =>-I :s;y:s; I Ava A.oya -l:Sx:Sl. H (2) naptcr'tavet 'tOV K"6KA.o acpou :fO=O. E�WJ..ou yta a = .!.2 TO (r1) f.xet flta A.u01] (k,m) ( 0, JO.) . H an6cr'tacr1J 'tOt> 0 an6 Ka9e eu9eia eivat = � · � , 01tOT£ TO {L) EX£t Tecrcrep� Mcre� (x,y) (ytati;) . ..J?{ . Av A.otn6v ..[� <.Jit. <1, Ol)A.aoi) � < a < 1 TeA.o� av 1 , 'to'te 11 ( 1) £xet ouo pi�e� e'tep6cr1JJ!£� 'tO't£ t>1tUPXOUV 8 A.ucre� (crx. 1 ) . Av a = 1 , unap ot onoie� <pucrtKa anoppin'tOV'tat. Apa 'tO (l:) eivat xouv 4 A.i>cret� (<JX. 2) , evro av a = � , t>1tUPXOUV aouva'tO. TeA.tKa yta va £xet A.i>crl) 'tO oUO'tl)J.l(1 1tpenaA.t 4 A.i>cre� (OX· 3 ). An6 'ta napanavoo eivat 1t£1 Kat apKei a e npo<pave� noo� av 0 < a < � , 'tO'te o K"6KA.o� ppi crKe'tat ev't6� 'tOt> n:'tpayrovou evro av a > 1 ('to 'te 8. Na i-..vOd -2 -4y=r -8, -y2 -4z=i -8, -t2 -4x=y -8 'tpayoovo ppicrKe'tat evT6� 'tOt> K"6KA.ou, ono'te 'to Ai>al) oUO'tl)J.la oev exet A.ucrl).] Me anaA.ot<pi) napovoJ.lacr'trov Kat np6cr9ecrl) Ka'ta J!SA.l) BXOU J.le: x4+y4+z4-8x-8y-8z= <:::>

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( 1),(2) => (x-z)[x2 +xz+:i +3y(x+z)+3y]=z-x=> {+ => (x-z)[(x +yi +(y+zi +(x +yXy+z)+ l] = O ( 1) ( k + m ) -2km=a fta x+y=k, y+z=m 1) ayK"6A.1) S yive'tat: k + m=l 'teA.tKa 'tO km= 1 a va £xet A.i>crl) (k,m) k�O, S=k2 + m2 + km + l = � [2k 2 + 2m2 + 2km + 2 J = ; m�O. ft' au't6 npenet Kat apKei 11 e�icroocrl) = �[( k + m)2 + k2 + m2 +2] > 0 . Apa x = z . 1-a-= 0 ( 1) va exet 't� p"':le� 'tl)� 1!11 apVT)- AvaA.oya X = y 01t0't£ 8x 3 = X => X = 0 i) t2 -t + 2 nKe�. H e�icroocrl) au'ti) exet OtaKpivoucra =2a - 1. .; =� =>x=y=z=O i) x= y=z=±.J_. TeA.tKa J.lovaot8 2../2 Av /1<0 Ol)A.aoi) a < .!_2 eivat aouva'tl). Av a�.!_ Ki) aKepata A.i>crl) eivat 11: (x, y,z)=(O,O, O) .

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1:.: op8oyrov10 Tpiyrovo ABr

( A = 90° ) dvat

AB=8, Br=lO. K6KJ..o� Ktv'Tpot> 0, not> �ttpxs Tat an:o TO r Kat E«pU1t'TETat <J111V n:J..&t> p a AB <J£ 01]pdo E, TEJlVEt 111V n:M:t>pa AB a£ 01]pdo A, rooT£ va dvat 2rA=AA. Yn:oJ.oyicns 111v an:o cnaOTI TOt> 0 an:o TO A.

R J5 · MN = R . 3 R � MN = 3 J5 2

E=MA=

4+2 2

=5 (=R), ea eivat Kat �0=5 .

IIW..t lffi rr.e cr'to MO�, �picrKOUJ.le MO= .J24 . Me Ilia aKOJ.lll XPTJ� 'tOU rr.e. cr'tO AMO, �piaKOUJ.le OA=7. B' Tp61to�: Bpi)Kall£ R=5, 61to'te

A�) = { OA )2 -R 2 =>Ar ·M=OA2 -5 2 =>

=> 6·4=0A2 -25 => 0A2 = 49 => 0A = 7

AO'KT) O'll 2 H aKTiva OT svo� flptKt>IU.iot> �taptTpot>

AB=2R, Eivat Ka8.:111 a111 «hap.:Tpo Kat To M d vat TO ptao 1'11 �· H AM Ttpvst TO 1]ptK'6.U.10 aTo N. 0sropovps 111v n:popoJ..'Il N ' T01> N <J111V AB. Yn:oJ..oyicn£ Ta TJlllJlaTa MN Kat NN ' , 0'1>VapT'Il a£t TOt> R.

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4R . . OM: -_ NN' ... � OM 'AN ' N ' exouJ.te. , -NN' _ 5 AM AM B ' Tpono�: 'ExouJ.te OBNM eyypa\lltJ.lO:

A-

=>AM·AN=AO·AB=> R2.fs AN=2R2 =>AN= if55R .

Tpono� (Movo pE n. 0Effip1]pa) : ' 'Ecr-rro MN=x, ON =y.

T6-re

RJ5 A M � 2 = J5 => J5 y = 01tO'te 2 ' y AO R 2 AN2 = AN '· AB => { x+AM )2 = { R+ y) 2R => RJ5-)z = { R+ y) 2R => J5 y+ => (l 2 3R K.A.1t. => 5 y2 +2Ry -3R2 =O=> y =5 =

A<JK1}<J1} 3

Tpiyrovo pE n:J..Et>pt� AB=lO, Ar=17,Br=21, d vat £YY£Ypapptvo a£ K'6KJ..o (O,R). a) Yn:oJ.oyi cns To V'I'O� AA TOt> Tptyrovov p) Yn:oJ.oyiaTE 1'11V aKTiVa TOt> K'6KJ..ot>.

Me II.S cr'to AOM, �piaKro 6n AM= R J5 . 2 'Ecr'tro T' 'tO avnotalffi'tptK6 'tOU T, 011A.aoi)

R + R = 3R .

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A1t6

Aua11 A ' Tpon:o�: Me II.S. crro Tpiyrovo ABr, �pi aKOUlffi on Ar=6, cruve1troc;; �r=2. E1tetoi) To KEv'tpO 'tOU JC6KA,ou �piaKe'tClt 1tUVffi cr'tT) lJ£crO Ka8e'tO Tou �r -ovoJ.tasro M To J!Scro Tou �r-, 11 01toia Cl1tSXet a1t6 'tTJV AB a1t6cr'ta� 0 -

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Ma9qpanKa yt.a TIJV B' A1>K£iou

Au<Jll Ax6 t11 ytviKW<J"'l 'tou n.e. £xouJ.ts: 1 7 2 = 1 02 + 2 1 2 - 2 · 2 1 · B� => B � = 6 => A�=8.

Ec;auou �Y = 2 Rucx

�Y

2 ucx

10 · l7

85 2·8 8 =

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AGKllGll 5

:Et Ki>Jdo (O,R), dva1. &yyqpapptvo UJo nM:t>po Tpiyrovo ABr. Av :r. dva1. <J11 p&io TO'U 2 Ki>Jdo1>, o&i;T& oTt: :EA1+:EB +:r.r1=6R1 • A

AGKllGll 4 . 01. oulp&GOl AM KUl BN &VO� op8oyrovio'U Tptyrovo'U ABr( A = 9Cr ) TtpvoVTat Ka8na aTo :r.. a) Bp&iT& TO Myo

J3 • y

Jl) A&i;T& OTl TU TpijpaTa rN,r:r.,rM pnopo'6v

VU U1tOT&i..tGO'UV 1tM:\>p� TplycOVO'U TO 01t0l0 · pa AlG'TU 8a dvat o;'U'Yrovto. r

Au<Jll

Me VOf.lO O"UVT1f.ltt6vrov crto AOB, naipvou� Ei>KoA.a 6tt a=Rf3 . 'Ecrtro N to avtt8ta�tptK6 tou A Kat M tO flSO'O tTJ� Br. An6 e . .1.taf.l8crrov O'tO 3R 2 :EB1, eivat: :EB2 + :Er2 = 2:EM2 + -

1°

R 2 => Of.toiro� crto :EON: :EN 2 + R 2 = 2:EM2 + 2

AUCJ11

2 a2 1 "' 2 = � = - AM = - a => � . Em-, 9 3 3 2 crll � : B:E2 = � BN = 4 2d + 2y -If = 9 3 4 2a2 + 2 r 2 - /3 2 = . AUa r 2 = AL2 + B :E2 = a) E ,

tvat:

A "'

( )

+ +2r -P 3tJ_ 3d +2r -P .... .:.... 31 .:...____;_____:. ._ _ => 9

12 =

5 r 2 + 2{32 9

R2 . Apa :EA 2 + :EB 2 + :Er2 = => 2LM2 = LN2 + 2 3R 2 = 2 + 2 2 = 6 2 R2 + = :EA 2 + :EN 2 + 4R R R 2 2

AGK11<J11 6 H AB &ivat otapnpo� tvo� Ki>Jdo1> Kat To r

tva <J11 pdo TO'U. :Et 'T1>X«io <J11 pdo A nt� otapt Tpo'U (j)tpro Ka8&'T1) npo� amijv, 11 onoia Ttpv&l 'T1)V Ar G'TO P, TOV Ki>Jdo G'TO I:. KUl T1)V 1tpot KTU<J11 nt� Br mo T. Ati;T& OTl A:E1 = AP ·AT A

=> 4r 2 = 2/32 => � = .fi . y

fta tTJV U1tOtsivoucra tropa txou�: a = rJ3 . :Euvsxro�: IN = rfi , IM= r../3 , r:E = 2:EA r 2 2 (:EA 8uif.tscro� crto opeoyrovto tpiyrovo :EAB � u non:ivoucra AB). A<pou r:E > rM > rN yta va anon:A.ouv auta nA.rope� tptyrovou apKei Au<Jll r:E < rM + rN, ii < rfi r.J3 , ii 2 < .fi f3 Enet8ii to tpiyrovo A:EB, eivat opeoyrovw, tcrx6et: r 2' 2 �:E2 = M · �B . ApKei A.om6v M · � = .1.P · .1.T , 2 , ii 4 < 5 + 2J6 nou tcrXt}et. E�<lUou 1M 2 :rN = T), �A = �T , 1tOU tOXUEt , ano·, tTJV Of.tOtOtTJta t(J)V �p �B 2 r 2 3r 2 5 r 2 "' , , - + - = - > Y 2 = r 2 , 01tOtE tO tptyrovo opeoyrovirov tptyrovrov A�P Kat T.1.B (exouv 4 4 4 A = T - O"Uf.l1tAT)prof.tattKe� tTJ� i8ta� yrovia� B). ea eivat o�rovto. B)

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B ' AYKEIOY 1\. CJKTJ OTJ

a , � "flU TU l<JXV01>v: a = (1, 8- a ii ), � =<2, 1 � 1 > a) Na i£1X,9Ei OTl: I � I = .J5 KUl a ii =s 1

AiVOV't'Ul TU (havVCJJ.LUTU

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p) Na 1>1tOloyta9d lJ yrovia ( a, � )

'Y) Na anoi£1X,9Ei OTl 1tpO� ji

a=�

i) Na avaA.1>8Ei To iu1V1>aJ.La a a£ ivo Ka9n£� (Jl)VlaTcOCJ£� U1t0 T� 01t0lE� TJ JllU ElVUl

- =(4, -3)

napclllTJATJ no p

E) Av 'Y

va ypa<pd TO 'Y ro� "fPUJ.LJllKO�

(J1)Vi1>UOJ.LO� TIDV a , p

A1>aTJ

� =(2,

Js IPI => IPI 2 =4+� 1PI2 => 4 1Pr =20

ypaf..Lf.t.L K6 <ruVOUUOf..LO 'tOOV u, p (BA£ne en6�o epW'tTJf..LU). 'Ex_ouf..Le A.otn6v a =A u +f..L P <=> (1,3)=A(1,-2)+f..L(2,1)<::> A.2A++2f..Lf..L == 13 <:::> f..LA. == -1 1 Apu v1 = 1{2,1) = (2,1) , v2 = -1 ( 1,-2) = (-1,2) . E) ;y =A. a +f..L P <=>(4,-3)=A(1,3 )+f..L( 2,1)<=>(4,-3) =(A.+2f..L, 3A.+f..L) <:::> 3A.A. ++2f..Lf..L ==-34 <:::>(A,f..L)=(-2,3). Apu y =-2 a +3 p

roe;

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Aivovrat Ta J.LTJ J.LTJOEV\Ka otav6 aJ.LaTa a , P , y , "flU Ta onoia l 01>V 01 CJXECJE� :

<JXV

JaJ = � 1� 1 = � J r l Kat a -2 P +r = o

a) Na UlTlOAo"('i]OET£ OTl:

a p = - � Jar

- 4=> J pJ 2 = 5 => ! PI = J5 . Apa p) Na OE\;ET£ . OTt: a = - 3 � p = 2, J5 = { 2,1), on6Te y) At:i;Tt: oTt : i l a J-1 � 1 = J o) Av v = 4y-a va od�ETE OTl v..iy :::>2 a P =10 :::> a P =5. Apa a = (1,8 - 5) = {1,3) . 2sj aj - - _ap_ = 5 = ----; J2 . 5 =A1>CJTJ p) cruv( a , � )=: ;: 1 a1 . 1� 1 v 10.v 5 5v2 2 a) a -2 P + r �o => a -2 P =- r => l a - 2 P I = I -r l 2 = 2 a - 2P r = 2 => 1t a 2 => P lrl 1�-r1 ( I l Apa ' ( a,� ) ="4 y) 'Ex_o1>�: 1tp0�a/ !p, on6Te umipx_et A.eiR, rocne 1 �2 -4( CiP) +41 �2 = 1 �2 => 1 �2 -4(CiP) +4(% 1 � = npo�P a = A.p . E�UUou a p = p npo�P a => = (% 1 a i => aP = - � I ar (1), a<pou U1t0 un66eCJ11 5= P CA. P )=> 5=A. P2 =>5=A. I PI 2 =>5=1... 5 => A.= I . Apa 1tp0�p a =1 P = P = { 2,1) . €xm> f..Le : ! PI = � l a! (2) o) Av V I ' v ot �TJ'tOU�ec; <ruvtcr'trocri::c;, ea E�auou: (2)=> J ai i PI = J al (� l al ) = � l ar (3) tcrX(>ouv; 2a = V1 + V , V1 j_ V , V1 // p . BpTJKU� p) 2 2 Apa: (1),(3)=> a p .; - l a! I PI =>a t-1- P , V1 = 1tp 0� p a = p = (2,1) , 01tO'tf: v = a - v1 =(1,3 )--(2 ,1)=(-1, 2) =>a =A. P f..Le A. < o=> J ai = I A. PI => � I PI = I A. I I PI => 2 ' T p ono� B llpo<pavwc; p ..i u = ( 1,-2), a<pou � - u = O. 1�.,1 =i => A. = -.± => a = -.± p 3 3 3 Apu v //u ' 01tO'tf: f..L1tOpou� vu YPcl'JIOU� 'tO a 2

( Js )

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EYKAE� B ' 89 -r.l/47

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------- Ma.fh}pa.nKa "f\O. TfiV B' A1>Ktio1>

l <il - 1 �1 � I UH I <il � l� l <il � 1:1 ( I ) AGKTJ (fTJ 4 -1 - = �-a+ ( - 3 -a)I = l l a-� = l al (2) a+P AivtTO.t T£Tpaxl.£upo ABrA AB =3 a+� ' , 4 4 4 ar =3 a +s � , =a +3 � , oxo1> a , � P.TJ ( I ), (2) � l al - 1�1 = la + �I · Anooeucv6etat Kat p.TJOtvtKa otav6ap.aTa. 'EniD M TO p.tao TO'U AB KO.t K TO p.tao TO'U Ar. aJ.I£cra a<pou a t..i � (Seropru.ta). a) Ati;Tt OTt TO. (fTJp.da A, M, r dvat (f'l)Vf:'U9ttO.Kcl a) V Y � 25��, 41 �; 4Y Y a Y � Y � <i 2 1 21t , va ppdTt: -) A (.:. 1-� p) Av a = � = 1 Kat a, p = ,, J ( = 25ara 4 1 r-12 -a- -r = -ar- 415 rl2 - I = i) To yovOJU'VO Kr ii) T�v �) 2 1 ai 2 1 l2 AUGTJ a) Lto tpiyrovo AB� , AM 4 a eivat otaJ.1£cro<;. � ay ( % l J ) - I ( UY ) .0 �o. l\pa V Y 2s l <il 2 Apa: AM= � (AB + M ) .1\GKTJGTJ 3 = � (3a+�+a+3�) na TO. Otav6ap.O.TO. OA ' OB ' or t<JXl>O'UV Ot = 2 (a+�) . Eni<JTJ<; OA+20B-30r =o, loAI = Ioa l =1, lorl = �. Ar =AB+Br = a) BpdTt Tt)V <JXETtK'I] Ot(fTJ T(I)V A, B, r = 3a + � + 3a + s� =6 (a+�) ( ) : OA OB , 1) OA: or < 90' �) Aoii;-r< LUVt1tro<; Ar =3 AM , on6tt A, M, r <ruveu8ttaKa. - - - - - 1p) i) KL\ = M-AK = a+3P - 2Ar = AUGTJ a) OA +20B-30f=O� ( O\-OJ =2 ( or-oB) � a+ 3� - � 6 (a+�) =-2 a , Kr = Ar-AK = rA= 2Br�rA= -2rB � rA tJ rB . 1- 1-a+P- ) � tO r ppicrKetat JJ£tu9> tOW B, A. ( Ar =3 Ar = =Ar2 2 p) OA+ 20B = 3 0r � ( OA + 20B r =9 0f 2 � Apa KL\ Kr =-6a (a+�) = 2 � l oAI +4 oA oB +4 1oBr =9 1orl � m>v 321t =-6-6 (-21 ) =-3 1-a,2 -6a- P=-6-6.l.l. 1' + 4 OA OB + 4.1'='1 ( � J => 1 ii) KL\ 1 -2 a l =2 1 a l =2 I -- �OA OB = O�OAl.. OB . 1Krl2 =9 (a+�r = 9 (1 al2 + 2a�+ l �n = y) OA+ 20B - 3 0f=O �OA-3 0r=-20B � ( OA-3 or r =40B2 � =9 [1+2.1.1 ( - �) +1] =9� 1Krl =3� !oAr - 6 OA Or +9 1 orl2 =41 0BI2 � KL\Kr = -3 = - � �m>V (KL\, Kr) IKL\ 2. 3 2 Kr I l I 1' - 60A or + 9 ( �J =-�.1' => (K-L\", Kr-) =-.21t3 � OA or = � �m>v {oA, or ) = <

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Al1KT/l1T/

E cn:w Ta p.ova6tala 6tavUl1p.a-ra a, p, y TfTOta WlTTE

(a, P) = i

ICat

(li, ) = � . y

l a - p + r l ;;::: I« + xp + Yl · 2 Av l1q: 'ExouJ.t£: 1a - p + rl ;;::: 1a + xp + rl �;:> (a - p + ji) ;;::: (a + xP + r/ �;::::> a 2 + p2 + ji2 - zap + zar - Zpji ;;::: a 2 + x2p2 + ji2 + zapx + zar + zxpji I;:> P2 - zap - zpr :::: x2 P2 + zapx + zprx I;:> 2 IPI - z 1a1 · IPI · (1Uv i - z iPI · I rl · lTVv i :::: x2 IPI 2 + ( z lai · IPI · l1tlv i + z lfil · lrl · l1tJV i) x � - ..fi ;;::: x2 + (1 + ..fi)x �;::::> x2 + (1 + ..fi)x + ..fi :::; 0 �;::::> (x + 1)(x + ..fi) so �;::::> x e [-..fi., -1] Al1KT/l1T/ Z: 'Ecn:w Tplywvo ABr . Av Aii = (6, -2), M = (3, 4) K«L H TO op96JC£VTp6 TO\J , va (Jp£9El TO Aii . Aim11 : 'ElTTW AH = (x, y) , E[vat Bf = M - AB = ( -3,6) -3x + 6y = O -3x + 6y = O =Z -3x + 6y = O H · Bf = O l. Bf AAA.a - =:> - =:> 0fi1 =:> =:> 3 + 4 = 0 =:> � - ' Apa AH = (Z,1) __.AB)__.Af x 1 1 0 H = 1,.nn Y AH · Af = AB · Af BH 1. Af BH · Af = 0 Al1Kfl l1fl 3 : Na V'TrOAoyturouv Ot ywv(E� TOV ABT av tlTxtJEt: (Ai M)Af + (M2 - AB2)AB = 0 (l) A = 9oo A = 9oo A = 90° AB . M = o AUGfi : A<pou AB t�Ar ea exou J.t£ : => => => - - AB2 = o 1-1 Ar 2 = 1 ---+ AB 1 2 ::::> 1 -Af l = 1 -AB I oF; = f = 45° Af2 Al1KT/l1T/ 4: AvOA = a = (-2, 4) JCat OB = p = (2, 1) 'dm: va fJpE8El 6tavvap.a y p.E I Y I = V4i) -rov o1ro lov o qJopt.a� va 6txo-rop.El TfiV ywvla -rwv OA JCat Oil. AUlTfl : 'El11'w y = (x, y). T6u lrl = ffl �;::::> x2 + y 2 = 4 0 (1) AAA.a ap = -Z · Z + 4 · 1 = 0 � a l. p , 07t6TE yta va OtXOTOJtE( 0 qJOpf.a<; TOV y TT/ ywv(a (a, p) ( .:r-.. 1t 1t ...-.. ' J. 1tp�1tE L K.at apKEl p , y) = 4 ( Z ) KaL (y, a ) = 4 ( 3 ) Na Av8El , (c)� 1rp0� X avll1Wl1fl

�

{

. -"-,- .

ExouJ.t£: (2) �l1Vv (p , y ) .:r-..

...-..

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{

../2 py ../2 (1) 2x+y ../2 . =2 �;::::> l7ii·I'YI = 2 �;::::> ,fS..f.W = 2 �;::::> Zx + y = 1 0 (1)

..[2

ay .J2 (1) -2x+4y ..[2 .. �;:>-x+2y=1 0 (n) =2 �;::::> ..;:w.rzo = 2 liiH'YI AJ.J..a (i),(ii)� x = Z, y = 6. (Ttjlf.� nou £1taAf16E'6ouv TTJV (1 )) . .1\pa ji = (Z,6)

Ojlotoo�: (3) �lTVV (y, a) 2 �;::::> •

Al1KT/l111

5: Av ABr tl161rAEVpo 1rAEvpa� a va fJpEBEl o yEwp.aptJC6� -r61ro� -rwv lTflp.Elwv M yta -ra o1rola

iM.i2 + MA · MB + MA · Mf = i a2 (l) Aul1f1 : IlapaTTJpOUJ.l£ 6n : ( 1 )� MA · (zM/i + MB + Mr) = i a 2 (2) Av 11 to !lf.cro Tll � Br, t6te : (2) �m · (zMii + zMJ) = i a 2 I;:> Mii(MA + MJ) = i6 a 2 (3) � · ZME - = 36 a 2 �MA -·Av E to jlf.cro tou All, t6t£ : (3) �MA ME = 323 a 2 (4) 1 Av Z to jlEcro tou AE, t6t£: (4) � (ZA - 'ZM)(ZE - ZM) = !i a 2 �;::::> (ZA - 'ZM)( -ZA - ZM) = : a2 �;::::> (ZM 2 9a2 �;::::> �a,f3) 2 �;::::> 1 3a �;::::> M c-(c) 6nou 71'\1711 ZM 2 - �-� ZA 2 = n3 a 2 �;::::> 1 ZM 1 2 = n3 a2 + (s ZM 1 2 = 64' L..tt J \.ZM + ---+ ZA ) = n3 a 2 �;::::> �-� ZM � = a a/3 A Al .= . jAZj = 41 1 � 3a TIapaTTJpl]OUJ.l£ on J. ' , Z Kat aKnva p = 8 ( c ) o KUKAO<; 8JlE Kevtpo tax{IEt

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Al1KT/l1T/ 6: Na 6EtX8EL 6n 6EV V'Trapxovv IC, A, p. E R TfTOtOt wurE ../IC - 1 + ../A - 1 + .,jp. - 1 > .,jKAp. + p. (l) AU lTfl : 'Ecrtoo 6n unaPXOUV K, ..l, Jl E R, (<pumKa K, A., Jl E [ 1 ,+ oo ) ) t&totot root£ va t<JXI}£t fl ( 1 ) T6t£ ../K - 1 + ../A. - 1 = .JK - 1 · 1 + 1 · ../..l - 1 = a(f. 61tou a = (.JK - 1, 1), p = (1, .r.r="') J.1£ lal = .../K, IPI _

= ..JI. 1a1 · IP I :::: ap , on6te : ( 1 )=> ap+.JJt - 1 > .,jK..lJt + Jt=> lai · IPI + .,/Jt - 1 > .JJt(K..l + 1)� + .JJt - 1 > ..[p.../KA. + 1. (2) Olloioo� : .JKX + .,/Jt - 1 = ...JKi. · 1 + 1.,/Jt - 1==¥8, 6nou ji = (...JKi., 1), 8 = (1, .[ii=T) J.t£ lrl = ../K..l + 1 , 181 = ..fP.', on6te : (2)=> y8 > lrl · 18 1 , npayjla O.tono

AJJ.iJ.

1 + ../A. - 1 + .,/Jt - 1 :S .,jKAJl + Jl. Ilpayjlan: ap :::; 1a1 · IP I =>v'K=l + ../A. - 1 s ...fK · ..JI = ..r;a ( 1 ) Kat r8 :::; lrl · l8 1 =>...fKi.+.Jjl=l :::; ../KA. + 1..[p. = .,jKAJl + Jl (2). M£ 1tp6cr6£<JT] tOOV ( 1 ), (2) Kata JlSATI 1t<llpVOUJ.1£ : .JK - 1 + ../..l - 1 + .JJt - 1 :S .,/KA.Jt + Jt (i) Ilapat1]pf1Gf1: Evt£Acll� av{J)..(Y'(a ppi<JKOUJ.l£ .JK - 1 + ../A. - 1 + .JJt - 1 :::; .,jKAJl + A. (ii) Kat ../K - 1 + ../A. - 1 + .JJt - 1 :::; .JKAJl + K (iii) Me np6cr6£<JT] toov (i), (ii), (iii) Kata Jill.TJ. naipvou J.t£: ../K - 1 + .JA. - 1 + .JJl - 1 :::; i (.JKAJl + K + .JKAJl + A. + .JKAJl + Jl) , llta ou<JKoA.t] avtcr6TTJta nou ea ll1tOpoucr£ va B 'Tponot; ApK£i va 0£�0\lJ,l£ 6n yta Ka6£ K, A, Jl E [ 1 ,+ oo) , t<JXI}£t fl .JK -

anOtf:Mcret f.va oEi>tepo epcllTTJil<l.

EYKAEI� B ' 89 T.l/49

r ' AYKEIOY -,

Ma9rt�JaTI Ka rtvi Kti� Tiallitia� Mt:xaA.11� Baat#..clKo�, lroaVV'Il� AaaKa#..01to1>A.o�, Krovmavrivo� l:epi(J)11� DapapTflpa E.M.E. KapoiTaa�

Aiv£Tat 11 G1>VclPTfiGfl f(x) = 1 - x 2 A. Na ppeOd TO 1t£Oio optapoil Tfl�· B. Na ppeOd TO 1t£Oio OfUGpoi> 111 � g(x) = f(lnx) r. Na A.vOovv Ol e;l(JID(J£� f(x) = 0, f(x) = 1 A. Na O£i;£T£ ern 0 � f(x) :S 1 yta KU0£ Tlp1\ T01>

x 1to1> ppiaK£Tat <no 1t£Oio optapoil Tfl� f

E. Na ppdT£ TfiV eA.a:xtGTfl Kat Tfl ptytGTfl Ttp1\

T<OV G1>Vap'f1\G£(1)V f, g

l:T. a. Na oti;ET£ 6Tt o#..a Ta Gflpda Tll� ypa(j)t t:xovv a1to Kl]� - 1tapa<rra«Jff � Tfl� G1>VUPTfiGfl� TfiV ap:x1\ T<OV a;ovrov ma0tp1\ a1to<rra «Jf1. p. llma dvat 11 ypa(j)tK'I] 1tapa<rra«Jff Tfl� y. Na ppeOoi>v ot Ttpt� 1t01> p1ropd va 1tapet o

ro<rr£ TO Gflpdo

(k,%)

ypa(J)tK'I] � 1tapa<rra«Jff � Tll � m>v O.pTfiGfl� f

A1tUVT11 Gfl A. fta va opi�E'tat 11

•

f(X) 1tpE1tEt Kat apKEi: 1-x2 0 (1) . 'E:xouJ.LE: (1) �x2 �1 �l xl2 �e � � l x l � 1 � -1 � x � 1 . 1\.pa Dr = [-1,1] B. fta va opi�E'tat 11 cruvapTTtOTJ g( x) = f (ln x) npenet Kat apKEi x 0 Kat x e D r . ME x>O e :xouJ.LE: lnx e Dr � -1�1nx � 1 � � � � ± � X � . 1\.pa Dg = [; J r. L'tO Df =[-1,1] E:XOUJ.LE: f(x)=0��1-x2 =0� :::::>1 -x2 =0�x=1, x=-1, f(x)=1��1-x2 =1�x=O A. llpo<pavro� f(x) 0 yta Kaee x e Dr . EniOTJ�, yta Kaee x e Dr E:XOUJ.LE: -x2 �0:::::> 0 �1-x2 �1:::::> :::::> O�.J1-x2 �1 :::::> 0� f(x) � 1 1\.pa 0 � f(x) � 1 yta Kaee x e Dr no 'tU r' 1tpOK61t'tEt on f(-l)=f(l)=O�f(x)�1=f(O) yta Kaee x e Dr o11A.aoi) 11 f e:xet eA.a:xtcrTTt nJ.LiJ TTtV f(-1) = f(l) = 0 Kat J!Eyt crTTt nJ.LiJ TTtV f(O) = 1 . AKoJ.La, yta Kaee x e Ds e:xouJ.LE: 0 � f(ln x) � 1 :::::> 0 � g(x) � 1 . ea E�E'tU<JOUJ.lE av unap:xouv X D g OO<J'tE: g(x) = 0 . 'E:xouJ.LE: g(x) = 0 � f(ln x) = 0 � �

e ---,

ln x

�

!:!

l:T. a. Ta OTJJ.LEia Trt� ypaq�tKT) � napacr'taOTJ � Trt�

f E:XOUV TTt J!Op<pi) M(x,f(x))= ( x,�1-x2 ) , xeDr Kat 11 U1tO<J'tUcrTJ 'tOU� U1t0 'tO 0(0, 0) EtVUt f; k (OM)= �(x-Oi + (�1-x2 -of =1 o11A.aoiJ cr'taeepiJ.

va dvat Gflpdo Tfl�

� In X = -1, In X = 1 � X = .!. ' X = . E1ttOT)� ea E�E'tU<JOUJ.lE av unap:xouv X Dg rome: g(x) = 1 . Eivat g(x)=1 �f(lnx)=1 � lnx =0�x=1. 'E'tcrt, g ( ±) = g( e) = 0 � g(x) � 1 = g(1) yta K6.9E X Dg 011MIOTJ 11 g E:XEt EM:XtcrTTt 'ttJ.lTJ TTtV g (±) = g( ) = 0 Kat J.LEytcrTTt nJ.LiJ TTtV g(1) = 1

p. fvropfl;OUJ.I.E On 'tU OTJJ.I.ElU 1tOU U1tE:XOUV U1t0 EvU maeepo OTJJ!EtO maeepi) U1tO<J'tUOTJ �pt(Jl(OV'tUt <JE K6KA.o J.lE KEV'tpo auto 'tO cr'taeepo OTJJ.LEio Kat a Ktiva TTt <J'taeepi) U1tO<J'tUOTJ. 1\.pa 'tU OTJJ!ElU Trt� ypa<ptKi)� napacr'tUOT)� Trt� <JUVUPTTtOTI� ea �pt(Jl(O V'tat cre K6KA.o J.lE KMpo 'tO Kat aKriva p Tipe1tEt va npocre�OUJ.I.E OJ.lffi� on ot 'tE'tayJ.LE VE� 'trov OTJJ.LEirov, 011A.aoi) m apteJ.Loi eivat m apteJ.Loi 'tou I:uvenro� 11 ypa<ptKT) napacr'taOTJ Trt� <JUVUPTTtOTJ� ea Eivat 'tO 11J!tlC61<A.to 'tOU JC6JCA.ou p 1t0U �pt(Jl(E'tat <J'tO 11J.ltEnl1tEOO _1tOU opfl;E-

0(0, 0)

f(x)

[0,1] .

( 0, = 1) 'tat U1t0 'tOV a�ova x 'x Kat 'tOV ee'ttKO 11J.Lta�ova Oy . ' (k,-53 ) UVTtKEt T0 OTJJ!EtO ' <JTTt ypa<ptKTt' 1tUpa-'

'Y•

<J'tUOTJ

Trt�

<JUVUPTTtOTJ�

e[-1,1])1-li = �5 �k= ±�5 �ke[-1,1], f(k)=��k 5 AGK11 Gfl

'Emro G1>VUPTfiGfl f : lR � lR 11 01roia dvat Mairo� ai>;ovaa <no ( -oo , 0] rv11 airo� (j)Oivovaa <nO

[ 0, +oo )

Kal 11 ypa(j)lK'I] Tfl� 1tapa<rra«Jff Tt

pV£l TOV x'x <rra Gflpda

y 'y <nO

( -2,0) , ( 2,0)

Kal TOV

(0,4)

A. Na ppeOd TO 1tp0Gflf'O Tll� G1>VclPTfiGfl�

B. N a O£i;£T£ ern 11 Ex£l pty\GTfl Tlp1\ r. Na T01t00£T1\G£T£ KaTcl ai>;ovaa G£lpcl T01>�

EYKAEIAHI; B' 89 -r.l/50

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«pl8p.o6�: 4, f(3), f(-1), f(2) A. Av 11 auvO:pTfl01) f dvat -rpuowp.o a. Na at:\;t:'Tt: O'Tl f(x) = 4 - x 2 ' X E lR p. Na oxt:atucrin TfiV ypaqnKl} Tfl� 1t«p0:m«CJ11 .

AmiVT110'11 A. E<p6crov T) ypa<ptldJ napacr'tacrT) TT)<; crovapTT)crT)<; OtEp:XS'tat U1t0 TU crl)�Sia ( -2, 0) , (2, 0) ea stVat: f( -2) = 0 Kat f(2) = 0 . Enstoi] 'l f sivat 'YVT)criro<;

au�oucra cr'to ( -oo, 0] ea EXOUf..l£ :

x < -2 => f(x) < f(-2) => f(x) < 0 Kat -2 < x < 0 => f( -2) < f(x) => f(x) > 0

Enstoi] T) f sivat 'YVT)criro<; <peivoucra cr'to [ 0, -too ) :

x > 2 => f(x) < f(2) => f(x) < 0 Kat 0 < x < 2 => f(x) > f(2) => f(x) > 0 T£/..o<;, a<pou 'l ypa<ptldJ TTJ<; napacr'tacrTJ TT)<; f OtEPXS'tat an6 TO ( 0, 4) ' ea sivat f(O) = 4 > 0

Apa f(x) > 0 yta Kci9S X E ( -2, 2) Kat f(x) < 0

yta Kaes x e ( -oo, -2) u (2, +oo)

B. H f sivat 'YVT)criro<; au�oucra cr'to ( -oo, 0] on6'tS: x � 0 => f(x) � f(O) . H f sivat 'YVT)criro<; <pei

voucra cr'to [ 0, -too ) , on6'ts: x 2 0 => f(x) � f(O) . Apa

yta Kaes x e lR tcrxf>st f(x) � f(O) on6'ts 'l f na poucrta�st cr'to 0 �E"(tcr'to 'to f(O) = 4 r. A<pou o apte�o<; 4 sivat 'l �tcrTTJ n�fl TT)<; cro vapTTJcrTJ<; ea 't01toes'tT)est 'tSA£1)'tUtO<;. An6 'tOU<; U1tOAOt1tOU<; 'tpSt<; 0 apte�o<; f(3) sivat apVT)TlKO<;, o f( -2) sivat 0 Kat o f( -1) sivat esnK6<;. Apa 'l ota'ta�T) 'trov apte�rov sivat: f(3) < f(-2) < f(-1) <4 A. Av T) crovapTT)crT) f sivat 'tpu:Ovu�o TOTS ea sivat f(x) = ax2 + bx + c, a ;t: 0 . Enstoi] f(O) = 4 ea npenst c = 4 . 'Exst pi�s<; 'tou<; apte�ou<; -2 Kat 2 on6'ts, an6 'tOU<; runou<; 'tou Vieta, 4 b c - = -2 + 2 => b = O Kat - = -2 · 2 => - = -4 => a = -1 a a a :Euvsnro<; f(x) = 4 - x2 , x e lR

------

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1'110 �u'J .; �u:m li o l.t1 .;

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Aivt:T«l 11 auvql}� auvUpTfl01) x3 - 3x + 2 f(x) =

x-1 a,

x*1 x=1

A. Na ppt:8d 11 -rtp.l} -rot> ap1.8p.o'6 a B. Na at:f;t:-rt: O'Tl 11 m>VUPTfiCJ11 dvat 1t«payroyi

<rlJ111 mo 1 r. Na 1Jpt:8d 11 1tapuyroyo� auvO:pTfiCJ11 Tll� f A. Na 1Jpt:8o'6v 01. op�ov-ru� t:<p«1t'TOJ1t:Vt:� t:t>8d t:� GTfl yp«<plKl} 1t«p«m«CJ11 Tfl� f

AnavT11 cr11

J2 -3x+2 (x-1Y(x+2) , Stvat f(x) = x -1 x-1 = (x - 1)(x + 2) E<p6crov 'l crovcipTT)crTJ sivat crovsxil<; exou�s: lim f(x) = f(l) => lim(x - 1)(x + 2) = a => A. rta X ;t: 1

x-+1

=> (1 - 1)(1 + 2) = a => a = 0

x-+1

rta h ;t: 0 sivat (1 + h - 1X1 + h + 2) - o h(h +3) h +3 h h h f(l + h) - f(l) = lim (h + 3) = 3 ' f'(1) = 3 Apa lim h-+0 h-+0 h r. na X ;t: 1 exou�s f(x) =(x-1Xx+2) =x2 +x - 2, f(l + h) - f(l)

01tO'tS f'(x) = 2x + l Kat s<p6crov, an6 'tO B . exou�s: f(I)=3=2·1+� ea StVat f(x)=2x+� yta Kaes X E JR . A. Ot opt�6vns<; s<pan't6�s<; crTTJ ypa<ptldJ napa cr'tacrTJ TT)<; f crs crT)f.!StO TT)<; ( x, f ( x)) sivat 01 su-

esis<; 1t0U EXOUV m>VTSAscr'tfl ol£69uVcrT)<; f'(x) = 0 .

1 AUa f'(x) = O � 2x +l = O � x = - Apa unapxst 2 �ovaotldJ opt�6vna s<pan'to� TT)<; ypa<ptldJ<; na-

pacr'tacrT)<; TTJ<; cr'to crTJf.!Sio TTJ<; , �· y= s-,tcrrocrTJ sxst AO'K1)01)

r ( -2) 1

(-�,r( �)) -

Kat

, T), y = -4

Aivt:T«l 11 auvO:pTfiCJ11 f(x) = aln x + �' x > 0 p.t:

a, � e lR

Av 11 t:<p«1t'Top.tvq Tfl� yp«<ptKl} � 1t«p«-

m«CJ11 � Tfl� f mo 01)p.t:io Tfl� p.t: -rov «;ova x'x yrovia

E. H ypa<ptldJ napama<J11 TT)<; cruvciPTTJ<J'Il<; sivat 'l

napa�ol..i] 'tOU napamivro crxft�a'to<; an6 TTJV onoia �OpOUf..l£ va S1taAT)erocroUf..1£ TU spro'ti]�'ta A.B.f.

4 A. Na ppt:OOOv Ol � 'T(I)V

( t,f( 1)} <JX11J1«'T�t:l

K«l Urn f(x) = 1 -ro-rt: x-+e

K«l 'TO map.sio 'TO ......� 'Til� ypa<puo)� 1rapuarama� 'Til� f J1t: wv x'x . ln (l + x) B. Na t>1toA.oyicrt:-rt: 'TO opto Urn .

EYKAEIAIIl: B ' 89 'T.l/51

a,f3 x-+0

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r. 'Emro

Ma&qpa'TtKU yta 'TfiV

M { x(t),f(x(t)) ) , x(t) > x(O) = 1

aq ,WO

7tot> IOvtlTUt <JTI)V ypaqma] xapcJ.maaq Tll� f roan: x' (t) = x(t) , cmot> t 0 Xf)OV� KlVI)<n)� Tot> aqf'do'U. '

0eropoof1£ Tll XPOIJoA.l] Tot> M mov y y Kat B TO aq,WO TOJ'll� Tll� ypaqnKI]� xapamaaq� Tll� f f1£ TOV x'x . Na ppdTE TO pl>9f'O JIETapoA.l]� Tot> EJ' paoot) Tot> TPM�iot> OBMA Tl1 aTt"ff'll xoo To aq ,WO OWf)lE'Tat a7tO Tl1 Otaq ( e, f (e)

A7tclVT11 011 A.

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A<pou 11 e<panToJ.ttvtl 'tTlc; ypa<ptKftc; napa<na<rrJc; 'tTJc; f <HO CJT)J.I.f:tO 'tTJc; ( 1, f ( 1) ) crxrJ J.I.a'tii;tt J.l.f: 'tOV a�ova x'x yrovia _::4 ea eivm: f'(l) = e<p_::4 = 1 E1tf:t0ft f'(x) = aX ea SXOUJ.I.f: f'(1) =1=> a1 =1=>a=1 AKOJ.1.11 limf(x) -+e = 1 => alne + p = 1 => p = 0 x Apa f(x) = In x, x > 0 H ypa<ptKft napacrTa<rrJ 'tTlc; CJUVclp'tTJCJT)c; 'tSJ.I.Vf:t 'tOV a�ova x'x <rtO OTJJ.I.f:tO (1, 0) a<pou f(x) = 0 <=> In x = 0 <=> x = 1 ln(1+x) = ln(1+x)-ln1 f(l +x)-f(l) B. 'EXOUJ.I.f:: X X X ln(1 f(1) x) f(1 x + + ) => lim = lim = f'(1) = 1 a0 -+ -+o x x x x <pou f'(x) = .!..X , x > O r. To 'tpane�to OBMA exet J.l.f:'YclAT) P«i<rrt tCJT) J.l.f: MA = x(t) J.l.tKpft j3aCJT) OB = 1 Kat U'JIOc; OA = f( x(t)) = lnx(t) .

r· Al>KdO'U

----

AaK11 011 3. A. Aivnat 11 m>Vcl PTTiall

l(x)=e·(1-x)-lnx, x>O. Na A.1>8d 11 E�iaroaq h(x) = 0 B. AiveTat ot m>vap'Tftae� f(x) =lnx+ax+l, x:>O Kat g(x) = e• + f3x, x e lR JIE a,f3 e lR

ypaqnKt� xapamaae� Trov ovo m>vap'Tftae rov txo'UV KOtVO <n)JittO TO A X0 , y 0 OTO 07t0t0

txo1>v TTIV iota KA.i<J11 . a. N a oei;ne OTt x 0 = 1 Kat

(

)

a - f3 = e - 1

p. Na OEigrE On 11 KOtvft E(j)MTOJ'ivll TIDV ypa(j)t KOOV xapamaaEIDv TIDV ooo m>vap'Tftaerov mo aqOWf)XE'Tat axo TllV aPX'l T(I)V �ovrov. ,WO

A( Xo'Yo)

y. Na ppe9o-6v ot Ttf't� TIDV

E7tt7tUOV OTt

a,p

ef" (e) + f' (e) = g" (1).

yvrop�O'UJIE

Ax6.VTTI<J11 A.

IIapa'tTJpOUJ.I.f: 6n: rta X = l t<JXI)tt h(l) = e1 (1 - 1) - In 1 = 0 rta X > l t<JXI)et 1 - X < 0 , In X > 0 01tO'tE: h(x) = ex(1 - x) - ln x < 0 na O < x < hcrXt}et 1 - x > O , ln x < O on6Te: h(x) = ex(l - x) - lnx > O :Euvenroc; T) e�icrro<rrJ h(x) = 0 exet J.I.OVaOtKft AUOTJ 'tOV apteJ.I.O X = 1 B. a. To OTJJ.I.f:tO A(x0,y0 ) eivm Kotv6 CJT)J.I.f:tO Trov ypa<ptKrov napamacrerov Trov f, g on6Te f(x 0) = g(x0) => lnx0 + ax0 + 1 = exo + f3x 0 => 1 (1) => a - ..., - exo -X }n X o - Xo o Eivat f(Xo)=_!_+a Kat g'(Xo )=eXo +f3. E<p6crov Xo cruvap'tftcretc; exouv 'tTJV iOta KAtCJT) (J'tO X o ea t<JXI)et: f(Xo)=g'(Xo) ::::1 :r- + a=e"o +f3=>a-f3= e"o --1 (2) Xo Xo exo -lnx0 - -1 =eXo - -1 => An6 ne; (1),(2) exouJ.�. e : Xo Xo To tJ.I.PaMv 'tOU eivat E(t) = .!_( 2 x(t) + 1)1nx(t) Kat => exo (1 - x - In x0 0 An6Xo'tO A. SXOUJ.I. E : X0 = 1 0) pueJ.1.6c; 'tOU J.l.f:'taj3oA:ftc; 0 x0 = 1 an6 'tTJV (2) naipvOUJ.I.E: a - P = e - 1 E'(t) = 21 (x(t) + 1), lnx(t) + 21 ( x(t) + 1)(Inx(t)), = na p. H Kotvft e<pa1t'tOJ.ttvtl 'tO)V cf > c (J'tO Kotv6 'touc; g x'(t) OTJJ.I. t io A ( x ,y 0), o11A.ao1] cr'to A(I, e + f3) eivm 0 = .!..2 x '(t)ln x(t) + .!_( 2 x(t) + 1) x(t) J.I.E y = Ax + J.l J.I.Op<pftc;: 'tTJc; 1 1 ' A. = f' (1) = g (1) = e + � ottA.aoft 'tTlc; J.I.Op<pftc;: = 2 x(t)ln x(t) + "2 ( x(t) + 1) . y = (e f3)x + ll . Enetoft TO A avftKet cr' auri)v e Otav 'tO CJT)J.I.f:tO M OtSPX,f:'tat an6 't1l ee<rrJ ( e, f(e)) xouJ.�.e:+ e + f3 = (e + f3) · 1 + J..l => J.1. = 0 . Apa 11 e�i crrocrft 'tTJc; etVat y = (e + f3)x , 1tOU 1tpo<pavroc;, OtSpx(t) = e ea eivm E '(t) = .!..2 e + .!..2 ( e + 1) = e + .!..2 • •

•

01.

EYKAEIAHI: B' 89 -r.l/52

------Eq>6crov 11 cruvaptT)<JT) h exet e"AiiXtcrtTJ ttflTJ tTJV a - a ln a tcr:x;Uet: h(x) � a - a ln a , yta Ka9e X E JR Kat a 01tOtoVOTJ1t0te 9ettK6 7tpUyf.!attK6. fta a = 1 1tpOKU1tte1 : ex :2: X + 1, yta Ka9e X E JR .

------- Ma9qp.a'Tt.Kcl yta 'T11V r A1>KEi01>

Xetat an6 tTJV apxfl tow a�6vrov. y.

1 1 f'(x) = � +a, g'(x)=ex +� Kat f''(x)= 7 , g''(x)=e" .

Apa

efw(e) + f'(e) = g'(l)<=:

Kat an6 tTJV I

a-�=e-1

{ e�)+� +a=e1 <::::> a = e

exouf.!e

�=1 .

\ I 0 \" 0 T O \" l (J J( (Jl (I K fHlTUTII ()ll\'(J fJT lJ crlE �11: T lJ Jl o!'J (ll: w nJ .; rr u p uy tiJyo tJ .

AO"K11 0"11 1 .

AivtT«l 11 m>vap'Tll <J11

h(x) = e<n a 2 x, x e JR. , -

a 6tTlKO� 1tpayJ1«TlKO� apt6J10�.

A. Na JU)..tTJ1ant TllV m>vap'Tll <J11 ro� npo� Tl1 JlOVOTovia Kat va ot�nt 6Tt £xtt tMIXlO"Tll TlJ11\. B. Na od�tTE OTl yta a = 1 11 tMlXlO"Tll TlJll\ Tll� m>vap'Tll <J11 � yivnat JlErlO"Tll ·

a 1 , yta In a � -a e ' � X + 1, yta Ka6t X E JR.

r. Na Otlx6o'6v Ol UVlO"OTllTE�:

Ka6t

a E ( O,+oo )

K«l

Anav•110"11

h'(x) = ae<XX - a2 = a ( e ax - a ) ' X E lR Kat In a , h'(x) = 0 <::!> e ax = a <::!> ax = In a <::!> X = a In a h'(x) > O <=:> x > -- , a In a h'(x) < O <=:> x < -a LUVe1tcO<; T) h eivat YV11 <Jiro<; q>Sivoucra crto ouicr'TT) ln a ' .e 1: ' flU � YVIlcrtOO<; au..,oucra crTO oUl<JtT)flU A. Eivat:

AO"K11<J11

napayroyl}, (at Tovou�), tvo� ytropytKo-6 npo ioVTo� O"t XPOVO X ano TllV tvap�11 Tll� Kallttp ytUl� TOt> UKOAot>6ti Tl1 m>vap'Tll <J11

�+ I �+3 f(x) = A - e 6 - e 6 , x e [ O,a]

onot>

a>O

xpovo� Mov onoio 11 1t«payroy1} Oa J1110tvlO"Tti. A. Na t>1tol...oyuno6v m � TIDV napaptrprov A, a B. Na ppdTt TOV XPOVO O"TOV onoio 11 napayroyl\ Oa £xtt Tl1 JlEylO"Tll TlJll\ Tll�· r. Na od�tTt OTl 0 pt>6J10� a-6�110"11 � Tll� napa yroyl}c; , MO otaO"Tll Jl« not> at>TJ1 at>;avtTat, m> Vf:Xro� JltlOOVETUl. •

A1tliVT11 0"11 A. Le XPOVO

i<JT) flB

X=0

TJ napayroyi) ea eivat 1tpoq>avroc; ...!?+3

0 , OT)"Aaoi} f(O)=O=>A-e 6

-ff'�+I =O=>A=e'l +e

•

• •

(

-oo ,

[�, In a ) +oo

]

,r � Kat napoucrta..,et e��. axtcrto crto <JT)f.!Bto

( )

ln a ln a = a - a In a flB ttflTJ h a a B. 'Ecrtro T) cruvaptT)<Jll f(a) = a-alna, a > O

11 onoia eKq>p�et tTJV e"Aaxt<JtTJ ttflTJ TT)<; cruvaptT)<JT)<; h . Eivat f' (a) = 1 - ln a - 1 = - In a, a > 0 on6te,

f'(a) = O <=:> ln a = O <=:> a = 1 , f' (a) > 0 <::::> In a < 0 <::::> 0 < a < 1 , f'(a) < O <=:> a > l .

•

• •

B. Eivat: Kat • • •

eivat YVIlcriro<; au�oucra crto ota<JtT)flU

( 0,1]

[ 1, +oo ) Kat na poucrt�et f.!Bytmo yta a = 1 flB ttflTJ f ( 1) = 1 r. Eq>6crov TJ cruvaptT)<JT) f exet f.!BrtcrtT) 'TlflTJ 'TT)V f{1) = 1 tcrx;Uet: f(a) ::;; 1 => a - a ln a ::;; 1 => a-1 In a � -- , yta Ka9e a e ( 0, +oo) a

YVIlcriro<; q>Sivoucra crto otacrtT)flU

a a =>e6 =1 it e6 = e2 =>a=O i) a = 12 => a = 12 -�+3

�+I

f(x) = e3 + e - e 6 - e6 , x e [0, 12] 1 -� +3 1 �+I f'(x) = -e 6 - -e6 , x e [0, 12] , on6te 6 6 2.+3 !+I X X f'(x) = O � e 6 =e6 �-- +3 =- +1 � x = 6 6 6 � f'(x) > O x e [0, 6) , f'(x) < O � x e (6, 1 2] .

H cruvliptTJ<Jll napayroyi)c; eivat YVIlcriro<; au�oucra crto otacrtT)f.!a [ 0, 6] , YVIlcriro<; q>Sivoucra crto [6, 1 2]

Kat napoucrui�et f.!BrtcrTo crtov 6° XPOVo an6 'TT)V evap�11 TT)<; KUMlepyeta<; tOU . f. fvropi�OUf.!B Ott TJ napayroyi) UU�UVetat crtO OlU1 2.+3 1 !+I <JtT)fla [0, 6] fle pu9fl6 au�TJ<JT)<; f'(x) =-e 6 -- e6

EYKAEIAIIT. B ' 89 T.l/53

------ Ma«huulnKO. yw TfiV

KcieE

X E [0, 6] 11 �ffil f' fu etVat yvr)crfro<; cp6i

VOOOa on6te o pue� a691mt<; nJ<; napayffiYll<; ,.tetro vetm cruvexffi<;. O'tO Oulcrrrt J.I.O. 1tOl) ClU't"iJ �avetaL 1.

AiVtTat l1 cn>VclPTll<Jtt

(a) Na <>ti;tTt oTt

)

a. = 1 Av g(x) = -- , x > 0 , TOTt va <>tl;tTt cnt ot x+1 X

cruvapn\at�

f ( X ) = X lnx, X > 0

(a) Na Jtd..tTttOd 11 f ro� xpo� TttV JlOVOTovia Kat Ta aKpOTaTa. (p) Na <>ti;tTt oTt u:rxvtt: xlnx + e-• � 0 yta KaOt x e O,+oo

(

g(x) = -- , x > 0 x+1 (p)

A <JKtt<Jtt

r· A1>Kdo1> ------

f, g dvat rvttairo� av;o1>at�.

(y) Na

<>tl;tTt

r 2o13 2014

OTt

2o14 ) ( ) r ( 2015

g( 2013) < g(2014)

Kat

A'6att :

A E t<JXUet: f(O) = 1 � _!_a. = 1 � a. = 1 ( O,+oo ) , ex (x2 + 1-2x) _- ex (x- 1)2 01tO'tE, (R) E\'Vat f'(X) (()) Na ppdTt TO <Jtt JltlO Ttt� mo oxoio tt t-. (x2 + 1)2 (x2 + 1)2 <pa1tTOJ1EvtJ Ttt� tivat xapallttltt <JTttV w f'(x) =O� x =1 Oda ( 11 ) : y = 2x - 3 f'(x) > 0 � x E ( --oo, 1) u ( 1, +oo) . Apa 11 f eivat yvT)O"tro<; au�oucra mo lR . E1tiQ"11 <; f{x)= lnx + 1, x > O Kat (a) Eivat 1 > 0 yta Ka9e x > 0 . Apa Kat 11 g ( x) = g' 1 -,x > 0 on6'te (x + 1)2 f'{x) =x+1 eivat yvT)crtro<; au�oucra O"'tO (O, +oo) 1 f'{x) =O � x =.!_e f'{x) > O � x >e (y) A<pou 11 g eivat yvT)O"tro<; au�oucra O"'tO ( 0, +oo) ea t<JXUet: 0 < 2013 < 201 4 � g(2013) < g(2014) 1 �-1 f'{x)<O� O<x <-e � 2013 < 201 4 acpou' -' t<JXI)Et' -AKOJlll 2014 2015 � 1 � H cruvap'tll ffil f A.out6v eivat yvrJcriro<; <p9ivoucra cr'to Ka9e �> 1 Kat 11 f eivat yvT)O"tro<; au�oucra. Apa: (0, ±] Kat yvT)O"tffi<; aU�OUO"a O"'tO [±, +oo) Kat 1ta ytar ( 2013 201 4 ) . ) < ( r p0UO"lcl�Et EAUXtO"'tO O"'tO X = e-1 'tO 2014 3. 2015 f(e-1 ) =e-1 lne-1 = -e-1 Aivnat cruvapTtt<Jtt (fl) Eno,.Wro<; yta Ka9e x e ( 0, +oo) t<JXUet: f { x) = a.x3 - ( a. + � ) x2 + ( 2a. - � ) x + a.2 , a.,x lR xapaaTa<Jtt Ttt� f <>txtTat mo <Jtt f(x) f(e-1 ) � xlnx -e-1 � lxlnx + e-1 ol HJtdoypa<ptK1) flt TtTflll llEvtt x = 2 t<pa1tTOJ1EvtJ xapalltt (y) fw(x) = .!_ , on6n: yta Ka9e J?rw(x)-xf'(x) + f(x)= ltt mov x'x . Exi<Jtt � t<JX'6tt limf'(x) = 3 x-+1 X (a) Na fJptiTt Ta a., � . =x2 .!.-x·(l (p) Na fltJ.tnlant TttV f ro� xpo� Ttt flOVOTovia X n x + 1)+xln x =x-xln x -x + xln x =0 Kat Ta aKpoTaTa. (<>) fta va f:xouv 'tllV iota KA.imt npenet Kat apKei: ' f (x) = 2 . f'(x) =2 � lnx +1 = 2 � x =e , (y) Na cn>yKpivtTt T� Ttp.i� r ( e-2013 ) , r ( e-2014 ) on6Te TO O"ll JlEio ena<pi]<; eivat TO A(e,f(e)), 011A.a.oi] A(e,e) Kat e<pan•oJ1€vll dvat 11 (a) A<pou 11 e<panToJ1€vll 'tll<; O"'tO A(2,f(2)) (e) :y=2x+�. OJlro<;: A e ( e )� e = 2e + � � � = e eivat 1tapill11J.11 O"'tOV a�ova x'x , t<JXUet: oeK'ti] TtJlTJ acpou e -3, o 11 A. a.oi] ( e )� (11 ) f{2)=0 . 'ExouJlE: f(x)=3a.x2 -2(a.+�)x+2a.-�, 2. 01tO'tE f ( 2) = 0 � � = 2a. (1) . E1ttQ"11 <; EXOUJlE: e AivtTat cruvapTtt<Jtt f (x) = _l__ Kat tva <Jtt limf{x) x-+1 2 -2{ � + �)x + 2a.-�} 3 x�-+1a.-� =I=3 �(2)lim(3a.x x + a. fldo Ttt� ypa<ptK1)� Ttt� xapama<Jtt � To A(0,1). (fl) An6 TO crUO"'tll Jla Trov (I) Kat (2) �piO"KOUJlE (y) Na <>ti;tTt oTt: XE

x 2 f" ( x ) - xf' ( x ) + f ( x ) = O,

(a) A<pou

•

A'6att :

•

A <JKtt <Jtt

�

•

A.JJ.iJ.

A'6att :

A<JKtt<Jtt

EYKAEIAH� B' 89 T.l/54

------ Ma&qpa'ttKO. yw 'Tl)V

f{ x) = -x 3 + 3 x 2 + 1

a = -1 Kat l3 = -2 01t6ts, Kat f{x) = -3x2 + 6x . Apa: • f { x) =O � x = Q � x = 2, • f{x) > O <:::> x e (0,2), • f{ X) < 0 <:::> X E ( -oo, O) u (2, +oo) .

r ' A1>Kdoo -----fiE Tl1 Om.KI\ 'tt'TJ111 Jlivrt, ttm romt, 11 upt9J111-

nJO\ OtaqH>pa Totl Jll\KOU� TOU V'fl� 0� wto

TO tppuoov Tou vu tivut 11 d«xum� 01>varq. B. Av (s) 11 t<pU1ri'OJ1Ev11 Tll� cf aTO A KUt r '

TO 0'11J1£iO 07101> Uml\ TtpVtt TOV y y , TOTS VU fJp£iTt: (a) TI)V t�(O'(l)0'1) Tl)� ( 8) (IJ) TO spfJUOOV TO'U Tpt"(cOVO'U OAr At>0'1) : A . 'Ecrtro A(a, a2 ) JlS a > 0 11 Kopuq>� 'tOU tpt •

"fVTT � q>Oivoucra crs Kaetva MO •a

[ 2, +oo) ,

(

-oo ,

Kat

011A.a5� 1tU�St tomK6 eNi:xtmo mo

0 tO f (O) = 1 Kat t01tlKO J,ltytmo mo 2 tO f(2) = 5 . (y) E1tsto� 0 < e-2014 < e-2013 < 1 Kat crto ouimT} JlU [ 0, 2] 11 f sivat "(V11 crtro� at>�oucra tcrxf>st: r ( e-2013 ) > r ( e-2014 )

AO'K11 0'1) 4 .

AivtTUt 11 O"UVUpT1) 0'1)

f ( x ) = .Je•- 20t3 + e20t3- x

(a) BpdTt TO ntoio optapot> Tll�·

(p) Nu TIJV �O't'Tt (l)� npo� TIJV povoToviu KUt TU UKpMUTU · ' (y) Nu od�tT£ Mt, e - 2013 + e20 13- � 2, X E Dr (o) Nu

od�tTt

g( X) = (2 (X) - e2013-x

O'tt 11 O"UVUPTI1 0'1J OSV txst UKpOTUTU.

At>a11 :

(a)

yrovou nou �picrKstat crto 1° tstaptT}Jl6pto. A6yro cruJlJlStpia� tou crxflJla'to� sivat B( -a, a 2 )

' 0 = a2 . ' tOU StvUt Kat tO U'JIO� Av OVOJlUO'OUJlS H (a ) 'TT}V apteJl11TI� otaq>opa tOU U'JIOU� U1t0 to EJl�UOOV 'tOU tptyrovou, exouJlS:

AB) (O�) 0 2a a2 - ( �) = · - a2 = a 3 - a2 2 2 Kat H(a) =li -2a=o(h-2) . Aq>ou a>O ea EXOUJlS: • H'(a) = O <:::> a = � • H'(a) > O <:::> a > � 3 . 3 � H'( a) < O<:::> O < a < . Apa 11 cruvap'tT}O'T} H(a) 3 2 , , StvUt "(V11 0't(l)� q> e'tVOUO'U <J'tO , 3 KUt "(V11 0't,-

H (a) = {

•

(o ]

H f exet 1tSOiO optcrJlOU tO D = lR aq>ou ' eY > 0 yta Kaes y e lR (l)� au�OUO'U O''tO +oo) . E1tO�(l)� 1tUpouex-2013 - e2013-x (p) 'ExouJlS: f{ x) = , o1t6ts: 2 2.Jex-2013 + e20 1 3-x , , ' /" 'I , O'tU�St 811.UXt<J'tO O'tUV a = 3 . 'EtO't, Ot KOpUq>S� • f ( X ) = 0 <:::> X = 2 01 3 • f ( X ) > 0 <:::> X > 2013 A, B tOU tptyrovou ea sivat ta O'T}JlelU • f{ X) < 0 <:::> X < 2 01 3 A � ' i Kat s -� ' i . Apa, 11 f sivat "(V11 criro� au�oucra crto [ 2 01 3, +oo) 3 9 3 9 Kat "fVll <Jl(l)� q>eivoucra O'tO ( 20 1 3 ] , SVOO 1tU B. («) H Eq>U1t't0� (s) crto A ea exst KA.iO'T} poucrtai;st sMixtcrto crto 20 1 3 to f ( 20 13) = J2 � = i , 01tO't8 (e) : y = i x + l3 , Kat A. = f 3 3 3 (y) Eq>6crov 11 (J'l)VCtptT}O'T} exet sA.axtcrto O'tO 2 01 3 �� tcrxf>st: f{ x) � f{2013) � .Jex-201 3 + e2013-x � .J2 � A e(s) <=> � = � · � + � <=> 13 = -� . Apa (s) : y= x 9 3 3 9 3 9 ex-20 1 3-x + e2013-x � 2, "(tU Kaes X E JR . e201 3-x = (o) g '(x) = 2f{x) f {x) + e x-2013 - e2013-x = 2 .Jex-2013 + e2013-x . .Jex-2013 e2013-x + e2013-x = + 2 e" -2013 = >0, yta Kaes x e lR . E1to�ro� 11 g sivat f

-oo ,

[%'

()

"fVll O'lro� UU�OUO'U O'tO JR KUt OSV 1tUpOUcrtai;st a KpO'tUtU. AO'K11 0'11 5. AivtTUt 11 O"UVUPT1J 0'1J f (x ) = x 2 , x e lR A. Nu qypa'flt'T£ O'TfiV nupwtav(l) nupupoA.l\ tvu tO'� Tp(y(l)VO OAB ' On01) 0 ( 0, 0 ) 11 upm T(l)V �OV(l)V KUt A fl KOpt>qnl TOU Tptyrovou

( )

�-----.

( or ) (M) = 'ExouJls ( OAr ) =

EYKAE� B' 89 T.l/55

4 2 9 3 4 =27 2

I 1

------ Ma.Oflpa.'t'tKa yta. 't't'lv r AuKtio'U

------

r ' AYKEIOY Aivt't'at TJ t;iaro(JT) : z 2 + P z + y = 0 (1) p£ p, '1 e R TJ onoia qtt pi�a 't'OV z1 no1> otv dvat 1\ CJ KT) CJT) 1

npaypa't'tKo� apt9po�. a) Na O£�£'t'£ O't't y > 0. p) AV Z z dvat T) allT) p�a 't'T)� napanavro t;iaroCJT)� 't'O't'£ va ot�t't'£ o't't o apt9po� W

= z t + Z z + yi . z t z z + P•

R av

a) E<p6crov 0 zl oev eivat 1tpay�anK6c; apt9�6c; yta tl)V OtaKpiVO'Vcra /). tl)c; ( 1) exou� /). < 0 � If <

p) Av

0 � 4y > If � y > 0. z1 , z2 eivm ot pfSec; tl)c; ( 1) 'tO'tf: a1t6 'touc;

ro1touc; Vieta eivat: z1 + z2 = -� = --13 Kat a

-z

1\ CJKT) CJT) 2

'ECJ't'ro z, w ptyaotKoi apt9poi '(la 't'O'V� onoio1>� t<JXiltt z = 2w + 9i . a. Av 't'a CJT)Jt£ia N w aVI]KO'VV CJ't'O Ki>Jdo p£ Ktv't'po K

( )

( 2, - 3 ) Kat aKTiva

, 't'M£ 't'a CJT)pda

M z o.VI]KO'VV £7tiCJT)� (J£ Ki>Jdo.

( )

dvat <paV't'aanKo�.

a. na 'ta OTJ�ia

� lz - 9i - 4 + 6il = 1 � lz - 4 - 3il = 1� lz - { 4 + 3i)l = 1

() C �f: KEv'tj)O 'tO crl)�io A ( 4 , 3 ) Kat UK'tlVU p= 1 .

p. ApKei va oe�ou� 6n u = -u .

z - 4 - 3i = <p,

o1t6te

ll = 1 � <P = .!. <p

�

N { w) exou�: lw-{ 2 - 3i)l = , ( 1)

Kat

1 --1 1 - <p <p - 1 <p - 1 - <p - 1 <p U = -- � U = =-- = -= -- = --- = -U <p + 1 <p + 1 <p + 1 _!_ + 1 1 + <p <p

'ECJ't'ro z , w e C WCJ't'£

•

(

w-z

)=4

•

1 z j:;t: 1

Na anood;t't't o't't: a. Av 't'a CJT)Jttia N ( w aVI]Ko'Vv (J't'T)V tlltUJITJ cl

Jt£

(

)

E 4, 0 ) ' E ' ( -4, 0

£CJ't'i£�

)

Kat

p'I} Ko�

peyalo1> a;ova iao pt 10, TOT£ Ta CJT)pda M(z) aVI]KO'VV CJ't'OV Ki>Jdo cl Jt£ KMpo 0 0, 0 ) Kat

aK't'iva p = 4 .

(

p. Av z 1 , z 2 dvat ptyaotKoi 't'COV onoirov 01

£tKOV£� aVI]KO'VV CJ't'OV napanavro Ki>Jdo, 't'O't'£

l z l - z 2 - 91 ;:;:: 1 .

y. Av 01 £tKOV£� A, B, r 't'ptrov ptyaotKWV apt9prov a, p, y dvat CJT)pda 't'O'V Ki>Jdo1> C2 roCJ't'£ 't'O 't'piyrovo ABr va dvat taonU'Vpo, TOT£ Kat 't'O 't'piyrovo 7t01> CJXT)Jta't'�O'VV 01 £1KOV� 't'COV u 1 =ap, u2=Pr Kat u3=ya, dvat tniCJT)� taonU'Vpo. AvaT)

z - 3i - 5 p. Na anoot�t't'£ o't't o apt9po� u = z - 3 1. - 3

AvaT)

� l z ;9i -2 + 3il = �

lw -(2 - 3i)l = �

1\<JKT)CJT) 3

, w z1 + 7- + yi = -P + ri = = r = y (2) 01tO'tf: . a y + Pt z1 z2 + Pt. , , <paV'tacrnKoc;. pe + 'Yi = i ( Pi + r) 1. 1tou etvat = -r + Pi r + Pi 2 1 2 1 2 1 2 1 y) A6yro 'tO'U (j3) eivat: w o 3+ w 0 4= i o 3+ i o 4= = i 4 · 503+1 + i 4· 503+ 2 = i + i 2 = i - 1 . o) E<p6crov o z1 = 2 - 3i eivm pfSa tl)c; ( 1 ) TJ allTJ pfSa tl)<; 9a f:lVUt 0 Z 2 = Z 1 = 2 + 3i Kat U1t0 'tOU<; ro1touc; (2) 1taipvou�: P = -4 Kaz y = 1 3 . Z1 · Z2

o1t6'tf: ( 1 ) �

0e't0u�e:

Au(JT) -4y

z - 9i z = 2 w + 9I. � w = -2

E1to�roc;, 'ta crl)�eia M z avfJKow cr'tov K1l�

£\Vat <paV't'aCJ't'tKO�.

2013 + 2014 w y) Na ppd't't 't'ov apt9po: w • p, y apt9poil� Na ppd't'£ 't'O'U� o) z1 = 2 - 3i .

Kat

N e C 1 � {NE) + {NE') = 1 0 � lw - 4l + lw + 41 = 1 0 , ( 1) 4 4 Kat z{w - z) = 4� w - z = - � w = z + - (2) z z a. 'Exou�e:

01t6'te:

�

�l

�

{1);; z + - 4 + z + + 4 = 1 0

EYKAEIAH� B ' 89 't'.l/56

------

Ma9qpanKu yta 'n)V r' AuJCdot)

2 2 Jr -4z+� + r +4z+� 10 j( z - 2) 1 + j ( z + 2) 1 1 0 I� l z -1 z -� - � I� 2 2 �lz -� + l z+� = 101� =:(z-2)( z-2)+( z+2){z+2)=1q� � ( z - 2 )(z - 2 ) + (z + 2 )(z + 2 ) = 1 0izl � zz - 2z - 2z + 4 + zz + 2 z + 2z + 4 = 1 O l zl 2 � l zl - 5 l zl + 4 = 0 � (l zl - 1 )( 1 zl - 4) = 0 _

lzl,ot

------

x 0 = 0 t1 Yo = -� x 0 • E1toJ,ltvco�, o �fltoUJ.1.8VO� 20 yt:COJ.1.8tptK6� t61tO� efvat Ot £U9£i£�: X = 0 (afJA. 0 21 cl�OVa� y ' y ) Kat y = - -X J.l.8 e):..,at'p€CJ11 tO. 20 CJ11 J.1.8{o 0(0,0) A.6yco t11 � (1 ) \'\ �

P. 'ExouJ.I.B im(z) = a Re(z) J.1.8 a e lR, , 01t6tt: .

1 f(z) = 5x + 2ax = x(5 + 2a) = 5 + 2a maeeP'tl x + axi x(1 + ai) 1 + ai ' '

•

l z l= 4 � M e C2 l. ApKei va ppo6J.1.8 to 0'6voA.o ttJ.I.IDV t11 � 2 p. Eivat: l z 1 - z2 - 9 1 � l l z 1 - z21 - 9 1 � 9 - l z 1 - z 2 1 m>vapt11 CJ11 � A. = 5 + � ( 1) afiA.aat1 yta 1rot£� nJ.tt� 1 + at �rooJ,ttvou 6n ta CJ11 J.t.Bia Mt, M2 avf)Kouv <Jtov tou A. e [ 0, +oo ) fJ e�i<JcoOfl ( 1) txet AUCJll eo� 1rpo� tc61CAo C2, BXOUJ.I.B: (M1M2 ) � 2 p = 8 � l z1 - z2 l � 8 � -l z1 - z2 1 � -8 � 9 - l z1 - z2 l � 1 . a e lR 'ExouJ.I.B (1 ) � l 5 + 2al = A.�1 + a2 "-· 2 2 2 Apa: l z1 - z 2 - 91 � 1 � (A. - 4)a - 2 0 a + A. - 25 = 0 1· Av A(a), B(p) Kat f(y), t6n: tCJXl)ouv: • Av �2. t6te yta va txet fJ (2) AUOfl 1tp81t£t Kat lai=IPI=Irl=4, (3 ) apKei � � O Kat (AB) = (Bf) = (fA) , (4) 2 AA.M : � � 0 � A. ( A. - 2 9 ) � 0 � 0 � a a =>

J.l.8 (4) �1 P - l=l 'Y - P 1=1 - 'Y I (5)

"'P .

fta t11V a1t6a£�t'J tou �fltoU J.1.8VOU, apKei va E�clA.A.ou t11V ttJ.I.tl A.=2 t11V 1taipvet yta a = . a1toae�ouJ.I.€ 6n I u2 - ut 1=1 u - U 2 1=1 ut - u I 3 3 TeA.tKa 0'6voA.o ttJ.I.roV t11 � Bivat to Eivat: l u 2 - ut l=l �y - a � 1=1 �(y - a) I Kat Jd'yt<Jt11 nJ.tt1 t11 � t1 = (3)

=I P II r - a l = 4 1 a - r l

OJ.toico� ppiGKOUJ.I.B 6tt I u - u2 I= 4 1 � - a I 3 Kat I u t - u 3 I= 4 1 'Y - � 1 . A1t6 t11V (5) txouJ.I.B: 4 1 P - a l= 4 1 r - P I= 4 1 a - r l =>l u2 - ut i = =I u - u 2 1=1 ut - u3 l 1tou eivat to �fltoU J.1.8VO. 3 AG KfJ<r11 4

4ivncn 11 CJUVUP'f11 cr1J f(z)=

z e C, z - 0 .

SRe(z)+llm(z) f1£ . z

a. Nu ppdt£ 1'0 y£<Of1£1'p1KO 1'0nO 1'<0V £1Kovmv 1'<0V fli'YU31KCJV Wen£ l f(z) l =2 Z

IJ. Av lm(z) = aRe(z) f1£ a e 1R 1'61'£: 1 . Nu 3et;He 6n 11 f(z) £ivu1 cnu8ef)1\ (uve;UP'f111'1) 1'01> z) 2. Nu p p£i1'£ 1'1) f.l£ylcn1J 1'1f1-q 1'01> •

'Eatco

A6GfJ

lf<z>l ·

z0 = x 0 + y0i J.1.8 x 0 , y0 e R Kat (X 0 ,Y0 ) * (0, 0) , (1), t6t£ a. lf(z0 )l = 2 � 5 Xo + 2�0 = 2 � l 5 x 0 + 2Yol = Xo + Yol 2�x 02 + y02 � � x 0 (2 1 X0 + 20y0 ) = 0

�

[o.�]u{ 2} [o.�J m.

AGKfJ<r11 5

4{VOV1'CU 01 f11YU31KO{ ICJXV£1:

Z 1 ,Z2 , Z YIU 1'01>� 01rOi01>�

l z1 1 = l z21 = l z3 l = 3 KUI

z1 + z2 + z3 =3 Nu 3£i;£1'£ 61'1: a) z1 + z2 + z3 = 9i -9 + 93 . z z2 z .

1 + 2z1z3 + 3z1z2 1 . lz1 2z2 + 3z3 l = -lz2z3 3 +

ziz2 + z2 z3 + z3zi = 31 ziz2 z3 . z 2 ) + Re( ...1. z1 z )> --3 . o) Re(- )+Re(2 z3 z2 z1

Al>GfJ

ICJXl)ouv:

1 � 1 = 1 � 1 = 1� 1 = 3 ( 1 ) Kat z1 + z2 + z3 =3 (2). a) AA.M: ( l )� l z1 1 = 3 � l z, l2 =9 � �� = 9 � -z, =9 z, 9,9 . A6yco t11� (2) o Kat 6J.tota: z3 = z2 = -

Z1 + Z2 + z3 £ivat 1tpayJ.tattK6�, 01t6t8 9a tCJXl)t:t: 9 9 9 + z2 zi + z 2 +

EYKAEI411I: B' 89 -r.l/57

z3 = zi

+z3 = - + - +

zi z2 z3

------1 (:::> -2 . = - =---2 . -

------- Mafh1panKU yw '"IV r' A1>KEW1>

= lz1 + 2 z2 + 3z3 1 = _2._ + .!! + 2 7 z1 z2 z3 l z1 + 2 z2 + 3 z3 1 = lz1 + 2z 2 +3 z3 1 = = 9z2 z3 + 18z1z3 + 27z1z2 _ l 9z2 z3 + 18Z1Z3 + 27z1z2 1 _ z.z2 z3 lz.z2z3 1 + 2 + 3z .z2 1 = 1 z 2 z3 + 2 ztz3 + 3z.z2 1 = 9 1z2z3 ztz3 9 3 · 3 · 3 z3 z z. 2 l l 1 1 1 l 1 = - l z2z3 + 2 z1 z3 + 3z1z2 1 3

fl) Eni<JTl �:

y) E;cW.ou

-.- 9 9 9 (2) � 2, +� �=3 � -2, +�+� =3 � - + - + -=3 � z. � 1-.3

ztz2 + z2z3 + z3zt = 3 z• z 2 z3 .

(2) � l z1 + Z 2 + z3 1 2 > 0 � ( Z1 + Z2 + Z3 )( Zt + Z 2 + Z3 ) > 0 � 9 9 + 9 0 9(z1 + z2 ( z1 + z2 + z3 X+- - )> � z1 + z1 z2 z3 z2 z3 z1 + z2 + � + � ) > -27 � z3 z1 z2 ()) 'EXOUJ.lS:

+�)+(� +� )+( � + !t.. ) >-3 (� z 2 z1 z3 z2 z1 z3 9 AIJ...a � = �· = z . = � K. t... n z 2 Z 2 1._ z1 z2

()

.1\aKfJ<JfJ 6

a. Na p ps8d o y£IDp£TplKO� T07ro� -rmv m)pdmv M(z) p£ z * -2 i KCll N ( w ) mm£ va laxUOt>V Ol

-l

(2 )

•

A6aTJ

z = x 0 + y0i

J.lE

67tOU cl alC'tiva p

- 2 = 0 (:::>

( )

1CUICA.o� J.lE KMPO K

(0 , - �)

Kat

= ; . OJ.tro�, TJ etK6va tou z=-2i, OTJI..aOft

cl . Tel..tKa o yeroJ.tEtptK6� t6no� sivat o 1CUICA.o�

to <Jt'lJ.lelO

cl :

•

A(O, -2) avftKet Cf'tOV 1CUICA.o

l z + � il = ;

xropi� to <Jt'lJ.lf:lO A.

i2007 =i4501+3 = -i,i2001 =i4502 =1 · 20 1 2 = 1·4·503 = 1 , 1· 20 13 = 1•4·503+ 1 = 1. 1

E;IDou eivat:

(2) <=> ; + 5 - l - wi + 5i l = 8 <=> lw + SI - 1( -i)( w -5)1 = 8 <=> lw + 51 - l-i llw - SI = 8 <=>l w -( -5)1 -lw -� = 8 <=>(NE') -(NE) = 8<=>N e (c;) . Onou c 2 eivat 0 lCA.fJ.oo� 'tTJ� U7teppoA.ft�, J.lS scrtis� E ( 5 , 0 } , E' ( -5 , 0) Kat crta9epft otaq>opa 2a 8 < 10 , nou ppiO'ICetat c:no TJJ.llf:7tf.m:oo nou ' opi�etat an6 tov a;ova y y Kat to aTIJ.tEio E. E7tl<Jt'l� y = 5 Kat a. = 4 , 07t6te P 2 = y 2 - a. 2 = 9 On6ts

Kat 11 e;i<JID<Jt'l tOU yeroJ.tEtptKOU t67tOU e{vat

x 2 y2 = 1 us> X C2 :- - �4 16 9 p. Av M 1 ( z1 ),M 2 ( z 2 ),N( w ) eivat ot etK6vs� trov z �, z2 Kat w avrlcrtOtxa, t6ts ta M1 , M2 eivat aTIJ.tEia tou 1CUICA.ou C 1 otaq>opettKa an6 to 3 A(0,-2) on6te l z1 - z 2 I = (M 1 M 2 ) � 2 · 2 = 3 , r-

� - l z. - z2 1 � -3 .

p. Av 01 p1ya81Koi ap18poi z1 , z 1 1Kavo1ro1o'6v Tfl CJXtm) (1) KOl 0 Jll"(031KO� ap18po� W lKOV07rOl£( T11 CJXtm) (2) Va 07r03£l;£T£ on: l z1 - Z 2 - wl � 1 a. 'Ecrtro

()

on6te: 2 Re( � ) +2 Re( �) + 2Re( � ) > -3 � z1 z2 z3 z2 + Re( z1 ) + Re( -) z3 ) > --3 . Re( 2 z3 z1 z2

crxto£�. z-i • e l , (1) KOl z + 2•. wi 1ooa + Si 1o t1 - wi 2oo7 + Si 2ot3 •

z-i z+i z+ 1 z- 1 iz <=> zz - 2iz - + 2e = -zz - zi - 2iz - 2e <=> 2zz - iz + iz -f 4i2 = 0 (:::> 2zz - i(z - z)-4=0 (:::> 2( X02 + y02 ) - i 2y0i - 4 0 2 1 9 � + Yo + 2 = 4 <=>MeC, (::> Xo2 + Yo2 + Yo

T6te:

x 0, y0 e R

Kat

z * -2i .

Eni<Jt'l� to aTIJ.tEio N avftKet c:nov lCA.fJ.o� 'tTJ� U7teppo/..ft � C 2 7tOU ppiO'ICetat atO 11J.llE7t£7tSOO

( y 'y, E) on6te l wl = (ON) � (OB) = 4 6nou B( 4, 0) eivat 11 (oe;ui) Kopuq>ft tl); uneppoA.ft� C 2

l:uvsnro�, txouJ.tE:

l z1 -Z 2 -wl � ll z1 - z 2 1 - lwll � fwl - l z1 z2 1 � 4-3=

7tOU eivat tO �11tOUJ.lSVO.

EYKAEidll}; B' 89 T. l/58

•

------

-------

Ma81UUlt\KU VLU tllV r' 1\uKatou

f ' AYKEIOY A9uvucno� KupupuyKlo#.u�, Krovmuvdvo� l:ep(cpfl�, 9ropu� l:Ko'6cpf1� llupupTilflU E.M.E. Kup&i'ftJU�

1\.GKTJGTJ 1

z20 1 4 = ( z 2 ) 1 007 . = 1 1 7 = 1· 4·25 1+3 = 1·3 = -1

Av Y'U 'fOV plyU&lKO Upl9po Z UJXUf:l l z - 1 1 + i l z l = 1 + 1 z1 - 1 1 'fM£ vu &e�ne o'fl:

a. z l = 1

�. z2 = -z

y. z7 = z

•

r� tpon:o<; (Eupt(fTJ

o. Av z t! R ,

l z l = l zi Kat l il = 1 on:6ts l z - 11 + i l z l = =1+il 2i-� 9z- � +il� =1+i l z- � Apa l z - 1 1 = l z l = 1

a. lax;t}et:

z = x + yi, x,y e R t6ts an:6 to a. 9a tax;t}st (x - 1) 2 + y2 = x 2 + y2 = 1 .Jj .Jj 1 ' ' z = 1 + 1. Apa x = ,y = ± lJ' ulJII.UulJ T 2 2 2 Av

'I - S:

(

]

.Jj z = .!. - i . Eivat z 2 = .!. ± i .Jj = 2 2 2 2 .Jj .Jj = .!. ± 2 . .!. . i - � = _.!. ± i = -z 4 2 2 4 2 2 y. An:6 to (p) tax;t}st: = - 1 (1) z 2 = -i => Z · Z2 = -Z · Z => Z3 = - l zl 2 (11) 'Etc:n, z7 = z3· 2+1 = ( z3 · z = ( -1 ) 2 · z = z . o . Ilapatf1 p01)f.1£ 6tt: ( 1) => z 3 + 1 = 0 => ={z+1) (i -z+1) =0=>i -z+1=0(2), aq>ou z t! R Kat lJ (2) osv sxst n:paypattd<; pi�s<; (d=-3<0). z46 = z3 '1 5+1 ( z3 z = ( -1 ys z = -z, E�UUou 2) z 1 = -z - -1 = --z 2 + 1 (=on:6ts z 46 + - = -1 4 z z z6 z

f\GKTJ GTJ

Av Y'U 'fOV fllYU&lKO upt9po

ltJXUf:l

l zl = 1

�z +- zz = 1 , 'fO'f£ vu &e�ne o'f\ z2014 = -1 (1 o� tpon:o<;)

l z l 1 => l zl 2 = 1 => z · z = 1 => z = .!.. z -1 + i = -(1 - i)(1 - i) = 1. => z2 = -2 1+i =

Kat

•

KU\

•

TOU

z = a. + Pi , on:6ts

Kat

( a.2 +P2 =1

A6GTJ

Ai> GTJ

E>ttouJ.ts

(l zl = 1

'fO'ff: z46 + 46 = -1 z

�·

) => 2a. = i) ( 2 + 2 = . => a. P 1 -2p1

�z +- zz = i

Kat

a.=P) => ·

a. = P = J2 => z = J2 ( 1 + i) => z 2 = i => 2 2 7 0 1 •100 0 14 0 z2 = ( z2 ) . = 1 7 = 1•4·251+3 = 1 3 = -1 •

1\.GKTJ GI'J

A. Av z t! R KUl

( .;.) z+

R 'fO'ff: vu &e�ne

O'ft: -l < z + - < l KU\ l z l = 1 1

B. Av z t! R KU\ z + ! = l zl , vu &e�ne m: l z l = 1 z

AUGTJ

z + .!.. = a e R. T6ts z + .!.. = a => z z z + .!.. e R => z + .!.. = a. => z2 - a.z + 1 = 0, a. e R. z z 0 apt9J.t6<; z sivat Mart tTt<; s�{Groart<; x 2 - a.x + 1 = 0 Kut aq>ou osv sivat n:paypanK6<; 9a

A. 'Emro

1tpS1tSt OtaKp{VOUGa d autft<; Va S{Vat apVlJttKfl Kat o8UtSpfl p�a tTt<; sivat o apt9J.t6<; z .

d<0=>a.2 -4<0=>-2<a.<2=>-2<z+-1 <2 z Kat an:6 tou<; tl'>n:ou<; tou Vieta z · z = ! => l zl 2 = 1 => l zl = 1 1 ' B. Emro z = x + yi f.1£ y * 0 . T6ts z + .!.. = l z l => x + yi + -1-. = �x 2 + y2 => X + yt Z y => x + yi + xx - i = �x 2 + y2 => 2 + y2 Apa:

EYKAEIAIIl: B' 89 -r.l/59

------- Ma&qpanKU Y14 'tfiV r' AuKdou

-------

Apa o yeroJ.I.StptK6� t61to� trov etK6vrov trov z eivat ellet'lfll J.l8 eat� ta CJTJJ.I.Sla E,E' ' Kat }li]Ko� JJEYaA.o'V a;ova 2 · 2 = 4 Kopu<p� tTJ� ellet'lf11 � KCX1 0 yeropnp1Ko.; -rono.; 'f<OV £1KOVIDV 'f(I)V eivat ta CJTJJ.I.Sia A(2,0),A'(-2,0) , ta ootoia anexoov p1ya31KIDV Z yta -rou.; 01tOiot>.; urxve1: an6 to 0 tT) JJEYaMtepq an6ataCJfJ an' 6A.a ta 1 1 CJfJJ.I.S{a tTJ� ellet'lf11�, Ka9ro� Kat ta CJTJJ.I.Sla lz - wj + lz - wj = 16 (1 ) . B(0, 1),B '(0, - 1) ta onoia anexoov an6 to O tfJ Ai> <fll E<p6aov w + w = 0 t6te 2 Re w = 0 oot6te o J.1.tKp6tePTJ an6ataCJfJ an' 6A.a ta CJTJJ.I.Sia tTJ� et'lfll �· 'Etm, ot }ltya5tKoi J.l8 to J.1.tKp6tepo w eivat <pavtaatuc6�. Av w = ai t6te t<JX(>et Sll }l8tpo eivat ot i, -i Kat autoi J.l8 to JJEYaA.mepo lwl = 2 => jail = 2 => j a j = 2 => a = ±2 . Ot n� too }l8tpo ot 2,-2 . w eivat: w = 2i , w = -2i A <f K11 <f11 6 A<f KTI<fll 4

w p1ya31Ko.; ap18po.; -rtrow.; m<ne w+w=O Ka1 l w l = l . Na ppe9o1'>v 01 npt.; -rou w

'E<nro

EuKoA.a ano5eucv6ooJ.1.8 6tt 11 e;taroCJfJ (1) yivetat: jz - 2ij 2 + jz + 2ij2 = 16 (2). An6 tTJV taut6tfJta 1 � + zl +l z, -zl = � � � 2 + � zl exooJ.I.S iz - 2ij2 +l z+2ij2 =16�21 �2 + � 2ij2 =16<=>j� =2 . tuvenro� o yeroJ.1.8tptK6� t6no� trov CJfJJ.I.SlroV M( z ) eivat o K6KA.o� J.l8 Kevtpo to 0(0, 0) Kat aJCtiva 2.

Eqp,eiwuq:

x0 + y0i rore: (2) <:> ... <:> x� + Yo 1 <=> M e ( c), 61rov (c): x2 + y2 1 .

IrolX&lrodcb�

Ol:raf.Je

8a f.J7rOpo6qaJ.Je va ppo6J.Je rov yewf.Jerpuco r61ro teal f.J& rov e'�� rp61ro: A v A(0, 2) teal B(0, -2) 01 &lteove� rwv

2i, - 2i avrf(ffolxa rore: (2) <:> MA2 + MB2 AB2 <:> AMB 90° <:> M e (c), =

(c): o te61CA.o� dlaf.JI:rpov AB.

A <fKTI<fll 5

'E<nro

61rov

p1ya31Ko.; ap18po.;.

l z - v'JI + l z + v'JI � 2.Ji ' Y1CX Ka9e p1ya31Ko ap18po z. A. Na 3£{9:-r£ On

B. Na p pd-re -ro yerope-rp1KO -rono -rrov £1Kovrov -rrov p1ya31Kmv z yta -rou.; onoiot>.; 1crxU£1

l z - v'JI + l z + v'JI = lvfJ ·

r. Na p peh£ 'fOV YE<OJIE'rp1KO 'fOnO 'f<OV £1KOV<OV -rrov p1ya31Krov z yta -rou.; onoiou.; 1crxU£1

'E<nro z p1ya31Ko.; ap18po.;. A. Na 3ei;e-re 6n

l l z - v'JI - I z + v'J I I S 2 vfJ

y1a

Ka9e p1ya31Ko B. Na p pd-re -ro yeropnp1Ko -rono -rrov £1Kovrov -rrov p1ya31Kmv yta -rou.; onoiot>.; 1crxU£1

z l z - v�JI - I z + v�JII = lvfJ

r. Na p pd-re -rov yeropnptKo -rono -rrov e1Kovrov

z l l z - v'JI - I z + v�JI I = 2 (1). Av z1 , z 1 31'>o ano -rou.; J11ya3tKoV.; 1tOU £1tCXA119eOOUV 'fi)V (1) JIE Re(z1) Re(z 1 ) 0 vex 3ef;e-re on jz1 - z 1 j � l

-rrov p1ya31Krov yta -rou.; onoiot>.; tc:rxUe1

•

Au <Jq

Av E(�,O) Kat E'(-�,0) Kat M 11 etK6va tot> z t6te 2� l lz - �l - l z + � I I = I ( ME ) - ( ME' )I s ( EE') = To iaov t<JX(>et av Kat J.1.6vo av to M eivat CJfJJ.I.S{O trov 11J.t.teu9eirov Ex Kat Ex' , 6noo x'x o «;ov� trov npay}lattKrov ato }ltya5tK6 emne5o. tuvenro� o yeroJ.I.StptK6� t6no� trov etK6vrov trov z 6tav l l z - v'31 - l z + �� =2� eivat ot 11J.t.teu9eie� Ex icat Ex' . A., B.

ll z - �l -l z+ �II =2 �I(ME) - (ME')I =2 · 1 (< 2�) 1\pa o yeroJ.I.StptK6� t6no� trov etK6vrov trov z eivat 'fO JllKpMepo KCX1 1t0101 'fO JI£YCXAV'r£po ptrpo; 11 U1tep�oA.f] J.l8 ea� ta CJfJJ.I.S{a E,E' , Kopu<p� Ail<fll tTJ� unep�oA.f]� eivat ta CJfJJ.I.S{a A(1,0),A'(-1,0) A, B. Av F(fJ,O)Kat E'(-J3,0) Kat M 11 etK6va too Av z1 , z Wo an6 too� }ltya5tKoU� noo 2 tTJV (1) J.l8 Re(z1 ) Re(z ) 0 t6te ot z t6te l z-�l + l z+ �I = (ME) + (ME') �(EE') =2� enaA.118e'6oov < 2 etK6ve� tOt>� 9a �p{CJJCOvtat ae 5ta<pOp8TIKoU� To iaov t<JX(>et av Kat J.t.6Vo av to M eivat CJfJJ.I.SlO KM5oo� tTJ� unep�oA.f]�. To J.l.tKp6tepo tJ.t.ilJ.I.a noo too t}lf]J.t.ato� EE' etat, o yeroJ.I.StptK6� t61to� trov evrovet too� Wo KMaoo� tTJ� unep�oAT]� eivat to M 6tav lz - �l + lz + �l = 2� eivat to AA' . eu96ypaJ.I.J.I.O t}li]J.t.a EE' . 'Etat, j z 1 - z j � ( AA') = 2 . H ta6tfJta t<JX(>et CJtT)V 2 r. l z- �l +lz+�I =4 � ( ME) + ( ME') =2 · 2 (> 2� ) nepimroCJfJ nou z1 = 1, z = - 1 f] z1 = -1, z 1 . 2 2

l z - v'JI + l z + v'JI = 4

•

II0101 ano -rou.;

z txouv

•

EYKAEI4� B ' 89 1'.1/60

TPIJ\OriA EAPMOrO N ErrPAliJ I M O N TETPAni\EVPON A<l>I E POMA O'tOV M IQU E L

reropyw� K. TPIANTOI.

To ap9po 1tOU aKoA.ou9ei O'K01teU8l va avaoeic;et tva A.ty6tepo yvroat6 9800P11!l<l, 1tOU avac:ptpetal (I)� «To nEV1'aypappo '1'01> Miquel» Kat aA.l8'6911KB a1t6 tllV auAA.oyi} to>V I11aomtrov*. H a1t6oet;11 -rou 9eoof'll j.t«t0� auto6 yivetal K<X't<XVOll'tiJ j.t6VO j.ltaa a1t6 OUO clUa 9eoop�j.t«ta - 1tOU <X1tOOWOVt<Xl 81tia'll � O"tOV Miquel Ka\ (l1t0't8AoUV (l1t� ec:papj.tOYt� 't(I)V ww't'f)tCDV 'f(I)V eyypa'lfij.tCDV 't8tpa1tA.8'6prov. EuKat.pia avac:pop� O'tllV eu9eia '1'0\) Vuibert oivetal O'tllV 1tPOOt11 8<pap�. -

E<ll A PMOrH (I)

tt� 1tA.eup� Bf, Ar, AB tptyrovou ABf 9eoopouj.l8 roxaia earoteptKa 0'11 j.l8ia A1 ,B1 ,f1 avttatoixro� Kat ovoj.t�ouj.l8 c. 'c2 ' c3 'tOU� 1t8plyeypaj.tj.ltvou� KUKAou� 'f(I)V tptyrovrov AB1f1 ,Bf1A1 , rA1B1 avnatoixro�. T6te taXt}ouv: A) Ot K'6KA.ot C1 ,C2 ,C3 OltPXovtal a1t6 to ww O"ll j.leio, tatoo P. B) Av Ol j.l8t�'6 tou� 1tapclU11A.e� 11!ll8U9eie� Ax , By ,fz ttj.lVOUV tOU� c. 'c2 'c3 O''t(l 0'11 j.l8ia 6, E, z ' t6te ta 0'11 j.l8la 6, E, Z , P eivat auveu9etaKa. (Eu9eia tou Vuibert). AllOAEISH : A) 'Eatoo p . to

OeUtepo KO\V6 0'11 j.l8lo 'tCDV KUKAroV c. ' c3 . ApKBi va oetx9ei 6tt to P a�Ket atov K'6KA.o C2 � to aut6 Pf1B P A1f . A1t6 to eyyeypaj.tj.ltvo atov K'6KA.o A

tetpa1tA.eupo AB1Pf 1 txouj.l8 Pr 1B = PB1 A (1 ) , A

evro a1t6 to eyyeypaj.tj.ltvo atov K'6KA.o C3 tetpa1tA.eupo fA1PB1 txouj.l8 PB1A = PA1f (2). A1t6 (1),(2) 1tpOKU1t't8l tO �11'toUj.l8VO. B) (t) A1t008lKVUOUj.t8 1tprota 6n ta O"ll j.leia P, 6, E eiv(:u auveu9etaKa. ApKBi va oe�ouj.l8 6n 6P I /EP , 1\

� 6tt: 6Pf1 = EPf1 • A1t6 to eyyeypaj.tj.ltvo atov K'6KA.o C1 tetpa1tA.eupo P6f1A txouj.l8 6Pf1 = .!\Af1 1\

(3 ) evro a1t6 to eyyeypaj.tj.ltvo atov K'6KA.o C2 tetpa1tA.eupo PEBA1 txouj.l8 EPf1 = EBf1 (4). E1t8W� 1\

BE / I M wxUBl EBf1 = .!\Af1 (5). A1t6 (3),(4),(5) 1tpOKU1t't8l tO �1lt0Uj.l8VO. (u) Mtvet va a1tooe�OUj.l8 6tt to 0'11 j.l8io Z a�K8l O'tllV eu9eia E6P. ApKei va oe�ouj.l8 6tt wx6et: A1PZ+ A1PE = 1t .ITpayj.tan, eivat: A1PZ+ A1PE = 1t - A1fZ+ 1t - A1BE = 21t - (A1fZ+ A1BE) 1\

= 21t - (BfZ+ fBE) = 21t - 1t = 1t . 1\

E<ll A PMOrH (II)

Eu9eia (e) ttj.tvet � 1tA.eupt� Br , Ar . AB tptyrovou ABr ata 0'11 j.l8ia A1 ,B1 ,f1 avttatoixro�. Io-xUouv: Ot 1teptyeypaj.tj.ltvot K'6KA.ot C. , C2 , C3 , C trov tptyrovrov AB1f. , BA1f. , rA1B. , ABf avnatoixro� OltPXOvtat a1t6 to tOto 0'11 j.l8io, tatro S. (0'11 j.l8io Miquel). B) Ta Ktvtpa K ,A, P , 0 trov K'6KA.rov C1 , C2 ,C3 ,C avnatoixro� eivat 0'11 j.l8ia tou autou K'6KA.ou, tatro c4 . ( KUKAo� 'tOU Miquel ) . r) To 0'11 j.l8io S eivat 81tl0'11 � 0'11 j.l8io tou K'6KA.ou C4 • A)

AllOAEISH A) Eiva\ c. = (AB.r. )

, c3 = (fA.B. ) . 'Eatro

tO OeUtepo ( j.l8'tcl 'tO

EYKAE!Am B' 89 -r.l/61

a. )

KO\V6 0'11 j.l8lo 't(I)V KUKAroV

-------

To Bqpo �o� E�� -------

C1 ' c3 . l) A1tOOSUCvUOUJ.18 6n tO s avftKst atOV K'UKA.o c = (AB f) . Ilpo� tOUtO apKd va od�OUJ.18 6n tO " " tstpa1tA.supo SABr sivat S"(r'Pcl'l'lJ.10, it "6n SAB = srAI .IlpayJ.tatt, a1t6 tO eyysypaJ.1J,ttvO atOV K'UKA.o cl " " " tstpa1tA.supo SAr IBI 1tpOKU1ttSi 6n: SAB = SArI = SBI AI ( I ), sv<.O a1t6 to eyysypaJ.1J,ItvO (TtOV c3 " " tstpa1tA.supo SB1fA1 1tpoK'61ttst SB1A1 = srA1 (2). A1t6 {1),{2) to �lltouJ.tSVo.

2) A1tOOS\KvUOUJ.1S 6n to s avftKst atOV K'UKA.o c2 = (BAifl ) ' OllA. 6n tO SAIBrl sivat S"(r'Pcl'l'lJ.10, it " " 6tt: SAIB = sriA . A1t6 tO eyysypaJ.1J,ItvO atOV c3 tstpcl1t�po SBirAI 1tpOKU1ttS\ 6tt: " " " SAIB = SAir = SBIA (3). A1t6 tO eyysypaJ.1J,ItvO atOV cl tStpci1tA.supo SBifiA 1tpOKU1ttS\ 6n: " " SB1A = Sf1A (4). A1t6 (3),{4), to �lltouJ.tSVo. " " B) A1toOstKvi>ouJ.18 6tt to tetpa1tA.supo OKPA sivat eyypa'l'tJ.10 as K'UKA.o C4 , it 6tt: OPK = OAK . H 0\clKsvtpo� KP t(J)V cl = (SABirl ) ' c3 = (SfAIBI ) dvat J.1800Kcl9St0� tll � KOlvTJ� xopoit� tOU� SBI . H 0\clKsvtpo� OA t(J)V c = (SABf) ' c2 = (SBAirl ) sivat J.1800Ka9Sto� tll � Kotvft� xopoit� tOU� SB . H 0\clKsvtpo� KA t(J)V cl = (SABifi ) , C2 = (SBAifl ) dvat J.1SOOKcieSto� tll � KOlvTJ� xopoit� tOU� srl . H 0\clKsvtpo� OP t(J)V c = (SABf) , c3 = (SfAIBI ) sivat J.1800Kcl9St0� tll � KOtvTJ� xopoit� tOU� sr .

Kata auv81tsta,

OP .l Sf

KP .L SB1

}

" " => OPK = B1Sf

(5)

Kat

OA .l SB

KA .l Sf1

}

" " => OAK = BSf1 1\

(6).

AK6J.111, a1t6 to

eyysypaJ.1J,ItvO atOV K'UKA.o c3 tStpcl1tAsupo SBirAl 1tpOKU1ttSt: BISr = BIAir = riAIB (7), sv<.O a1t6 to " " eyysypaJ.1J.1SvO atOV K'UKA.o C2 tstpci1tA.supo Sf1BA1 1tpOKU1ttSt: f1A1B = f1SB (8). A1t6 (7),{8) 81tetat A

6tt B1Sf = BSf1

OPK = OAK , 011A.. to �lltOUJ.lSVO. " " r) fta va sivat to S aTJJ.18io to K'UKA.ou C4 = (OKPA) apKsi va t<JXI}st: SOP = SAP . IlpayJ.tatt, sivat A

SOP =

(9). A1t6 (5),(6),{9) 1tpoK'61ttst 6tt: A

· S0f = SBf = SAr = SAB1 = Sf1B1 = Sf1A1 = EYKAEIA�

· SAA1 = SAP .

B' 89 �.1/61

------

To 81\pu <to'U E1>KI..t:t6q ------

E<!)APMOfH (Ill) te 1CUpt6 1tEVta:yrovo ABrdE Kaee J.tia a1t6 � 1tA.Eupe� tou crxrtJ.tati�et tpiyrovo J.I.E t� 1tpoeKtaae� trov 1tpO<JKEt�rov <JE autft 1tA.Eupv tOU 1tEVtayvou. OvOJ.t�OUJ.I.E ABZ ' are ' rdH ' dEI ' EAK ta crxrtJ.tat�OJ.tEVa tpiyrova, cl ' c2 ' cl ' c4 ' Cs tOU� 1tEptyeypaJ.t�OU� KUKAoU� trov tptyvrov autv avnatoixro� Kat A , , N , P ,T ta oe6tepa Kotva <rll J.tEia trov IC6KA..rov autv avttatoixro�. Tote, to 1tEVtayrovo AMNPT eivat E'Y'YP«l'lftJ.to ae IC6KA.o.

ADOAEISH (I) l:to tpiyrovo IZf 11 eu9Eia ABE> Eivat otateJ.tvouaa. A1t6 tT)V aqulpJ&OY'I\ (11), ot IC6KA.ot C1 = (ZAB) , C2 = (fBE>) , C6 = (IZr) auvtpexouv <JtO iOto <J11 J.I.E{O ( 1tOU Kat O.Vcl'YKTIV Eivat tO ) A. (BJ.hte omA.av6 <JX!iJ.ta � ). Apa, TO «n)J&8W A avqK£1. mov K't)KJ..o c6 = (IZr) .

(11) l:to tpiyrovo IZf 11 eu9eia EdH Eivat otateJ.tvouaa. A1t6 tT)V acpapJ&f>111 (11), ot IC6KA.ot Cl = (dfH) , C4 = (ME) , C6 = (IZr) auvtpexouv mo ioto <rll J.tEio ( 1tou Kat ava'YKTlv Eivat to ) N. (BJ.hte Ot1tA.av6 <JX!iJ.ta � ). Apa, TO «n)f'£(0 N «V1\K8l mov K't)KJ..o c6 = (IZr) .

EYKAEIAHl:. B' 89 -r.l/63

- · -

C7 -

------

To Bqpa TO\l Ev1CA.d6q

(Ill). OvoJ.Ul�OUJ.lB S TllV tOfJ.TJ tcov eu9et.O>v TA Kat IN. ea 08�0UJ.1B 6tt to s eivat OT)J.lBlo tOU K(ncA.ou C = (TAN) . ApKei to SNAT va eivat eyypa'lftf.I.O, "

iJ 6tt: ANS+ ATS = 1t . Ilpay,...att, a1t6 to eyyeypafJ.J,.ltvO O'tOV K6KA.o c6 t8tpcl1tAeupo INAZ exouJ.lB ANS = ANI = 1t - AZI = 1t - AZA ( 1 ) . A1t6 to eyyeypafJ.J,.ltvO O'tOV K6KA.o cl t8tpci1tA.eupo A

AZTA exouJ.lB: AZA = ATA = ATS (2). A1t6 t� ( 1 ) Kat (2) 1tpoK61tt8t to �TJtOUJ.lBVO. Apa, 1'« cn)p&ia S, A,T, N a\'1]K01>V cnov K'6KA.o C . (IV) Ta OT)J.lBia E ,T, N avi)Kouv otou� <pope� tcov nA.euprov tou tptyrovou IAS. A1t6 TllV &<p ap pO'Y11 (1) Ot K6KA.ot c 4 = (IEN) , C s = ( ATE)

C7 = (STN) cn>V1'p£xoov cno Uho cn)p£io ( 1tOU Kat aVclYKTJV eivat tO ) P. (BMne omA.av6 axflfJ.a.) Apa, 1'0 P a\'1]K&t cnov K'6KA.o C = (SATN) . (V) - (VI). H rAI eivat OtatefJ.VOUO'a tOU tptyrovou ZEH. A1t6 TllV &<pappO'Y11 (11), ot K6KA.ot C3 = (M"H) , C4 = (ME) , C7 = (ZEH) cn>VTp£xovv cno taw cn)p&io ( 1tou Kat avciYKTJV eivat to ) N. (BAt1te omA.av6 axflfJ.a -+ ) . Apa, 1'0 cn)p&io N a\'1]K&l cnov K'6KA.o C7 = (ZEH) . H BAK eivat otatefJ.vouoa tou tptyrovou ZEH. A1t6 TllV &<pappoy1] (11), Ot K6KA.ot cl = (ZAB) C5 = (AEK) , C7 = (ZEH) cn>VTptxovv cno taw cn)p&io ( 1tOU Kat aVclYKTJV eivat to ) T. Apa, 1'0 cn)p&io T a\'1]K&I. cnov K'6KA.o C7 = (ZEH) . (VII). OvOfJ.cl�OuJ.lB L TllV tOfJ.TJ tcov eu9et.O>v AB Kat HN. ea 08�0UJ.1B 6tt tO L eivat OT)J.lBlO tOU K6KA.ou C = (TAN) . ApKei to TANL va eivat "

eyypa'lftf.I.O iJ 6tt: TNL+ TAL = 1t . IlpciyfJ.att, a1t6 tO eyyeypafJ.J,.ltvO O'tOV K6KA.o C1 t8tpcl1tAeup0 BAZT exouJ.lB TAL = TAB = TZB = TZH ( 1 ). A1t6 to eyyeypafJ.J..ttvo otov K6KA.o C7 tetpc11tA.eupo A

ZTNH exouJ.lB: TZH+ TNH = 1t Kat A6yco Tll � ( 1 )

1tpoK61ttBt 6tt: TAL+ TNH = 1t � TAL+ TNL = 1t (VIII) Ta A, r, N avTJKOUV O'tOU� <pop� tCOV nA.euprov tOU BLH. A1t6 TllV &<pappoy1'](1) ot C 2 = (BrA) K6KA.otc; = (HfN),C = (LNA) , cn>V1'p£xovv cno ffiw cn)p&io (nou Kat avciYKTJV eivat to) M. (ourA.av6 oxfl,...a .) Apa, 1'0 cn)p&io M av1']K£1. cnov K'6KA.o C = (LNA) . TeA.tKa, 6A.a 1'« T,A,M,N,P av1]Kovv cnov K'6KA.o C. 1\

• COURS DE MATHEMATJQUES ELEMENTAIRES EXERCISES DE GEOMETRJE COMPRENANT PAR F.G.M CJNQUJEME

6. rKIOKA. T. EllJMEAHTOY E.M. llOAYTEXNEIOY EKAOI:Eit KAPABIA • 1 952 EDITION

METAfllPAl:H :

EYKAEIAm B' 89 T. l/64

------

I I

... -,..,.;:Mi:::-...

, ... / T I

Dllel (11) .

------ To Bi)J1« Tot> E1JKAEi&) ------M l rAa i KO I AP0 M O I KAI rEO M ETPIA Eivat yvcoor6 6n f.UyaOuc� z=a+Pi avn O'tot:Xei O'tO OT)f.l£lo M(a,p) it O'tO OulvuOf.W OM = (a, p) -rou J.uya5tKo6 B1WtEOou at avrimpo<pa. E1tO,..S 11 .1\J.:yeppa -rro Jll'YaOtKOOv �etil.;etat f.l£ 't11V AvaA.'UTI.KI} at 'tl1 AuxvooJUmKI\ rerof.l£'tpia, O 01t0� npoocr� Tl)� Ev� rerof.l.B'tp�. A'U'tl) 'tl1 olQ0'1)vomq AvaAmucll� rerof.l.B'tp� .:1UIVOOf.W'tUdl� rerof.l.B'tp� EuKA.ei � rerof.l.B'tp� ea &oUf.l£ ma 1tpoPA.ftJ.W-ra 1tou aKo

Knee

v�

apteJ.l.(Ov

JltYaOtKrov aptef.IOOv,

A.oueoW. J !! fl P O BAHMA

apt.9p.ot)� z1

•

v K

clvat

•

apte�

, Z2 , Z3

rt.ropyo� TaanaKi011�

(01tOU 11 11 'tS'tap'tll KOpU<pTJ 'tOU 1tapaJ..\flOU 'tOU o1toiou ot 'tpet<; MA£<; Kopucpe<; eivat "ta O,A,B) S'tO't I'MI = 1 -& 1 = 1 & 1 = p , e1to J,Jivro<; 'tO 11 eivat O'll f.£.1 tO 'tOU Ki>KA.ou (O,p) . y

Kat

Av yt.a TOU� p.t.yaot.Ko'6� l«JltlO'UV :

l z t l = l zz l = l z3 1 = P > 0 z 1 + z2 + z3 = 0

o.:i;T£ O't't. ot. Et.Kov� Tou� G't'o p.t.yaot.Ko .:ni n.:oo Eivat. Kopu(J)t� uronA£6pou Tpt.yrovou.

rta 6A£<; 'tt<; a1tO�eil;et<; 1tOU aKA.oueouv ea 1tpS1tS1 01 ZJ, Z2 , Z3 va eivat ava �UO �ta<popsn KOi . Au't6 tcrxUEt yta'ti av TJ't«V 1t.X Z 1 = Z z , 'tO'tS z 1 + z 2 + z3 = 0 �2z1 = -z3 � 2 lz l l = l z3 1 � 2p = p � p = 0 , U't01t0. J !l A1to0£t�11 [JlE Aval..U't't.K'fl r.:rop.npia]

9S't0Uf..1£ : Z 1 = a1 +i l3t> Zz = a2 +i 13 2 Kat z3 = a3 + i l33, 01tO'tS 01 elKOVe<; 'tOU<; <HO J.nya �tKO emxeOo eivat 'ta A(av 13t), B( Uz, 13z) Kat r(a3, I33)E1tet�i] l z1 1 = lz 2 1 = l z3 1 = p > 0 , 'tO 'tpiyrovo ABr sivat syysypaf.LJ.Jivo <Hov Ki>KA.o (O,p). 'Exou fJ£ : z 1 + z2 + z3 = �

To OAI1B sivat p6f.113o<; ((OA)=(OB)=p), e'tcrt (OA)=(AI1)=(11B)=(OB)=p=(011). 1\.pa 'ta OM Kat OB/1 eivat tcr61tA.eupa. E1toJ,Jivro<; LAO!l = 6 0°=Lfl0B Kat e1tet�i] r011 euesia (M = -or) ea eivat LAOB=LBOf = LfOA = .:1.

1 2 0° . Apa A B r tcr61tA.eupo .

J!l A1toO£l�1l [AJ..y.:ppt.K'fl]

Eivat yYOOO''tO on yta Kaee Z I , Z2 E( tcrx6et: l z 1 + Zz l 2 + l z1 - Zz l 2 = 2 l zt l 2 + 2 l zz l 2 ( 1 ) (A01C11 011 9 oeA.ioa 1 0 1 oxoA.tKou) A<po6 Z1 + Z2 = -Z3 ea SxOUf.l£: ( 1 )=> 1 -z3 1 2 + l z1 - Zz l 2 = 2 l zt l 2 + 2 l zz l 2 => l z1 - Zz l = -./3p, a<pou

l z� l = l z z l = l z3 j = p .

Dfl01a: l zz - z3 1 = -./3p = l z3 - Zt l · .:1.

� Ur + � +0:3 + i(l3t +132 +133) = O �Ur +� +0:3 = 0 at + az + a3 = 0 Kat Kat l31 + l32 + l3 3 = 0 � 3

1\.pa (AB)=(Br)=(rA), A B r tcr61tA.eupo . 4!1 A1toO£t�ll [AJ..y.:ppt.K'fl] Eivat : l zt l 2 = l zz l 2 = l z3 l 2 = p 2 =>Zt Zt = p2 p2 p2 ZzZz = z3 z3 = p 2 =>Zt = -, Zz = -, z3 = -

13 . "tou ABr). Apa "to K.l3. "tou

'EXOUflS:

f3t + f3 z + f3 3 3

�

, xo=O Kat yo=O (o1tou G 'tO K. .:1.

A B r "tamfSe'tat f..1£ ·'to 1tEptKSV'tpO 'tOU; yt' aU'tO 'tO 'tpiyrovo Et Vat tcr61tA.eupo. 2'1 A1t00el�ll [p.E At.avuap.a't't.K'i) r.:rop.E't'pia] 'Emro A, B, r 01 elKOVS<; 't(OV Zt ' Z2 ' Z3 aV'ti O''t01X« O''tO fltya�tK6 em1te�o . Ta A , B , r si vat 0'1lf..1£ia 'tOU Ki>KA.ou (O,p) . 'ExoufJ£: z1 +. z2 + z3 = 0 � - - - - - � OA + OB + Or = 0 � OA + OB = -Or

Z1 + z2 + Z3 = 0 � Zt + Z 2 + Z3 � 2 2 2 p p p � - + - + - = 0 =>z1z + z z3 + z3z 1 = 0 ( 1 ) 2 2 zi z2 � 2 Es«Uou: Zt + Zz + z3 = o � ( � +Zz + � ) = 0 � =

( I)

�

� +z; +zi +2( z,� +�Z:3 +Z:3Z,) = � +z; +zi 0 (2) Eivat (z1 - z2 l = z� + z; - 2 z1 z2 = = -zi - z 1z 2 + z2z3 + z 1 z3 ( z2 - z3 )( z3 - z1 ) .

EYKAEIAIII: B' 89 T.l/65

------

To B1]pa TOll Ellldl:wq ------

A pa (z1 - z 2) = (z1 - z 2)(z2 - z 3 ) (z 3 - Zt) · 3 Df.lOtU (z2 - z 3) = (z 1 - z2)(z2 - z3)(z3 - z1 )

3 (z 3 - z1) = (z1 - z 2)(z 2 - z 3 ) (z 3 - z1 ) 'Etm l z1 - z 2 1 = l z2 - z 3 1 = l z 3 - z1 1 , OTJ I..aofl t.

A B r tcr61tA&upo. :EXOAIA: To 1tpOTJ'YOUf.I£VO 1tp6�J.:ruta eivat Klaocn

Ko OTO £ii)o� TOll Kat JCI)KAo<pOptt CJ£ apKe'tE<; 1tapaiJ.a ye<; 61teo<; ot: a. Av 01 ptya3tKoi zh z2, z3 dvat 3w<po ptTtKoi ava 3\>o Kat loxUO'UV • Zt + Z + Z3 = 0 • z1 2 + z 2 + z3 2 = 0 2 2 3£\;T£ OTl 01 £1KOV£� TCOV OTO ptya3tKO £1tm£30 dvat KOpll(j)t� to01tUi>poo Tptyvoll .

A1to()Et�11 2 2 A1t6 t� z1 + z 2 + z 3 =· 0 Kat z1 + z 2 + z 3 2 = 0 1tUtpVOUJ..LE z1z2 + z 2 z 3 + z 3 z1 = 0, 01t0tE U VU'YOf.lUcrtE <JtfJV 1tP01l'YOUf.lEVTJ 4:!1 U1tOOE�T). Mta aMTI AU<JT) Eivat: T) (z - z1 ) (z - z 2)(z - z 3) = 0 EXEt pfSE� tou� Z J , Z2, Z3 Kat ypaq>Etat:

- (z1 + z2 + z 3) z + (z1z 2 + z2z 3 + Z3Z 1 ) z ...... z1z2 z 3 = 0 � z 3 - O. z 2 + O. z - z1 z 2 z 3 = 0 � z 3 = Zt z2 z 3 ' 1tOU E1tUAT)eWEtat yta z = Zt ' z = z2 Kat z = z 3 . Apa z1 3 = z1z 2 z 3 z2 3 = ZtZ2 Z3 , z2 3 = ZtZ2Z3 3 3 3 E1tof.!Evro� Zt = z 2 = z 3 Kat etm l zt l = I z2 1 = I z 3 1 , apa avay6 f.lU<JtE crto 1 R 1tp6PA.TJ flU. p. Av Ot p.tya()tKOi Zt, Zz, ZJ tivat ava ()'\)o 2

z1 2 z2 z3

()taq)OpETtKOi Kat t<JX'60'VV = Kat = TOTE ot EtKOVE� T(I)VZt, Zz, ZJ cno p.tya()tKo EnbrE()o tivat KOptl<pt� umni.E6potl

z2 2 z1z3,

TPt1rovo'V

Ano()Et�TI A<pou 01 aptef.loi Eivat ava ouo Ota<popOt U1tO KAEtEtat va Eivat Ka1toto� a1t6 autou� icro� f.lE tO f.lT)OEv Ot6n tOtE ea UmlPXE Kat OWtepo� aptef.l6� icro� f.lE to f.lT)OEv, at01tO. Me 1toA.\f.l6 Kata f.lEATJ trov U1toeecrErov 1taip2 2 2 2 VOUf.le. z1 z 2 = z1z 2 z3 ::::::> z3 = z1z 2 ( a<pou •

•

z1z 2 * 0). Me a<paipe<JTJ Kata !lEA.11 trov 2 2 z1 = z2 z3 Kat z2 = z1 z3 1tUtpVOUJ..LE : z1 2 - zl = z 3 (z2 - Zt ) ::::::> ( � -�) ( � +�) =-�A � -�)

::::::> Z1 + Z 2 = -z3 (z1 * z 2 )::::::> z 1 + z 2 + z 3 = 0 ( 1 ) A1t0' � z1 2 = z2z 3 z1 3 = z1z 2 z 3 ' 2 z2 = z 3 z1 1tatpVOUf.le z 2 3 = z1z 2 z 3

z l = z1z 2 Apa z1 3 = z2 3 = z 3 3 Kat l zt l

z 3 3 = z1z2z3 l z2 l = l z 3 1 > 0 (

A6yro (1 ),(2) tO 1tp6PA11f.lU UVcl1etat crto 1 R

•

Zt 2 Z2 2 Z3 2

1 · Av ot f.lt1aOtKoi Zt, zz, z3 tivat ()ta<popnt = Koi ava ()1)0 Kat t<JX'6Et + + + + TOTE Ot EtKOVE� T(I)V Zt, Zz , Z3 CJTO f.lt1aOtKO E1tins()o tivat KOptl<pE� t G01tl..£6 potl TPt1IDVO'V

z1z2 z2 z3 z3z1,

Ano()st�TI E>etOUJ..LE : U = z1 - z2 , V = z2 - Z3 ; V!{ = z3 - Z t f..l£ 1tp6cr9EOT} KUta pSATJ 1taipVOUf..l£ U + V + W = 0. Eivat u2 + v2 + w2 = (z1 - z2)2 + (z 2 - z3 ) 2 + (z3 - z1)2 = 2 (z1 2 + z 2 2 + z l - z1z2 - z2 z3 - - z3z1 ) = 0 a1t6 (a) •

l ul = I � = l wl ::::::> l zi - z2 1 = l z2 - z3 1 = I � - z� l ::::::> A B r

tcr61tA&upo. 2!! llPOBAHMA [2!! 0. llToi..Ep.aiotl)

I:s KaOs TtTpanAE'Vpo ABrA, ()s�Tt 6Tt AB·rA+AA·Br>_AT·BA

Ano()st�TI E>eropOUf.lE tOU� f.ll1U01KOU� aptef.lOU� Z I , Z2, Z3 , Z4 1tOU exouv etKOVe� t� KOpU<pE� A, B, r Kat !1 avticrtmxa tou tetpa1tA.Wpou. Eivat AB · rl1 + M · Br = l z t - z 2 l l z 3 - z4 1 + l z t - z4 l l z 2 - z 3 1 ;::: I Czt - z 2)(z 3 - z4) + (z1 - z4) (z2 - z 3) 1 . . . = = •

I �Zz - �z4 - Zz� + �z4 1 = I � - � II Zz - z4 1 = Ar · & .

I:XOAIA: • H U1t00et� 'tOU 2!!!! e . TI'toA.epaiou pe EuKA.eiO&ta J.ticra J11tOpei va yiv&t pe opotO'tTJ'ta 1) pe a V'ttmpo<pl). TIO:V'tro<; 11 a1tOOet� pe Mty®tKl'J f&rope'tpia &ivat mo a1tA1) a1t6 'tt<; 1tpOTJYOUf.I£Ve<;. • To icrov mo cru J11tepacrf.1a 'tOO eeropl'Jpa•o<; t<JX6et, O'taV 'tO 'te'tpcl1tAeupo eivat eyyp6.\jftf.10 (1!! e. TI't0Aef.1aiou). H a1t60eti;TJ 'tOU crTTJ pil;;e'tat cre yrovtaKe<; axecret<; 'tTJ<; Mty®tKl)<; ferope 'tpia<; 1tOU &ivat UATJ «£KTo� T1)� i>A.q� T1)� r! AllK£ioll». J!! llPOBAHMA [0tp.a EMll Av ot

1959) :

EtKOVE� T(I)V f.lt1aOtKIDV apt9p.rov Zt, Zz, Z3, ()ev tivat GUVE'VOttaKa CJ1lp.tia Kat t«JXVO'VV:

• l zt l l z2 l l z3 1 = 1 Re(z1) Re(z2 ) Re(z3) l zt - z2 ll z2 - z3 11 z3 - Ztl ::5 4 •

TOTE

;::: 0 ,

;::: 0

Kat

;::: 0

l:KE'JIEt�: - Tz fJeJ.ovp.s va &i�ovp.s; l z t - z 2 l l z 2 - z 3 l l z 3 - Zt l ::5 4 - lid)(; p.craax'f/p.ari?;craz yswp.crpzKa 11 npo'fJ yovp.eV'fJ axeCJrf ; apy::;;4 , 01tOU a , p , 1 Ot 1tAEupE� tOU tptyffi VOU f.le Kopu<pe� t� etK6ve� A , B ,r trov z1 , z2 , Z3 crto f.ttya<itK6 e1ti1tt<io .

EYKAEIAHI:. B' 89 T.l/66

L91P· 68

d 01 AO>A9�D A0>1 l}Xdn Sro 31'fno��mg AD lDN

.' J3BlilOllOY:ll:o VII. J3dOllrf 5roll�rf

' u>�oyJtgyoll YJ13>ldv tvt1.J3 lw?XD Ut1.3rfgol.uodll H-

· l z _ £z

Wz - "z ll "z - Iz l <

< I Zz

- z l l lz - z i i Z z - tz l + l tz - z W z - z l j lz - E z l + I E z - z 1 1 z z - z W z - Z z l : lD13cbl)dA. lp,nn 3191 ' nXto1.0JlAD J tmt s:'v' d n1 S3A9)lt3 31i S<,tO)ltgnA.trl Sno1

Ez ' <:z ' lz ' z 31ino.ol)rlOAO AV

.' II.CfJ>Ltqvl.trf lw?XiJ 30 V?:vtt3qOllV

t1.Ul 3rfaotli?d1Vl3rf VII. 3rfgodollrf 5cpiJ - :St31h;nt::I

' .IHV 0.0.1. Aq>d<t3"(lt A<O.l. IDII).ri D.l. tDAJ3 l'd'D O.Olt9 '

'A. 'D < Hd 'V d Vd ' Jd J d ' Hd D 3.1.9.1. '.I8V <lOA -q>A.td.l. o.o.1. aog�utua 0.0.1. OJ3rilw tDAJ3 d AV

·A. +

· fJ

· fJ +

V:WHVH O dll S o · A(j)Tiijtdn A(j))ltgl>A. -tti O<).!? SJ;Idodmtg Slu. nod1?Tf no1 l>}3Altrld3 lpttd13ti(l}3A lu.o tn13]}dlu.o 1)l)I1Wnwno nrllt"(�<;>dlt o� : oiVO X':I

. {\,Qi\(!)A.1d1 {\,01 0}3Tilw 9)l1d31(1)�3 lDi\}3 0 01 ndv; ·n<Y'()l<pt no1 0}3Tilw o 01 lD)l ( d')I) no"{)[ no1 l})l1d31oo.o3 mAJ3 ..1 ' s: ' v n1 SooA?tfoltg: -<p

�1 �

I bu)

I bm)

�I I

' I > = I- 1- I = I > £ = - �- I = z 1 > 1 z 1 = 11 - z - 1 1 = (}IV) : mAJ3 ·og31t}lt3 9)l1gnA.1ri OW nX10W}1i\'O E Z - 1 ' z Z - 1 ' Z - 1 i\(1)1 S3i\9)l13 10 J ' s: ' V (1)1.0? ll))l I =d Di\}1-)l'O lD)l ( I ' 0))1 01 Od1J\?)l 3Ti O"()l<pl i\01 3Ti <;tOd003 @

a: ::IHVI:JV}L\3:

S\1 ntA. AV .'�Xotd33t rmi 9JlV ro;? � VD?ri 1Vti.J3 OJ3riUo VII.? ti.V 3riCJOOII.VdJOJlV VII. 3riC)OdOJlri 5!3DCfJJJlld33t 53to3l 3:J otqrlotijltAno tX<} nrllt"(�<;>dii noAIJ)A.td1 S<;>A3 C})ltd31(1}�3 ow tn13)1!)Jd� OJ3rllto n�q uo : 3riao;J3f? v11. 3riaor?e '.1 - : St 3 lh?)( ':I 0

'9:L<lD OArolJd.l. 0.1. 9ltD (I)�� tD.1.3)(0Jdlf A(I)A9�D

A(l).l. I}.XdD h 3.1.9.1. 'og3lt}lt3 9)(tgDltri 0.1.0 <lOA -q>A.td.l. S�cbo.do)( tDAJ3

S3A9)(13 to AV ' I >

lzl

E z - I ',z - I 'z - I Aro:a. 'z :;9)(tgnA.tri row':!(,

3ri

V:WHV90dll uf' no}.td31:i Sltlttil)lltW o 30(1}!?? Alu. norl So1nrli)."(�Od1t no1 ltoltAIJ)cb)l3 All� • A(j}rl()tdn A(j))ltgt>A.tti nrllt"(�<;>dlt 30 Sn}d13ti(l}3j Snt3!?J3"()1n3 Slu. nrllt"(�<;>dlt A1J11}. nolt nrllt"(�<;>dlt 6z 01 IJ)A3 'S'O}d13ti{l}3J nt3!?J3"()1n3 Slu. nrllt"(�<;>dlt 30 A(j}rl()tdn A(j))ltgn.ttti So1ntii)."(�Odlt S<;>A3 l)..too.tnAYJ 3tinA1)l)l 91nn nrllt"(�<;>d1t 013: • ! V I V O X':I 0

. ., 5 A.g n 3191to ' I 5

�t ndv; A.gn z

I= I · Z->J-V ·V)I - 5lN ·V)I -5ZV·..E -5 I I I I z

5 vv ..IS: 1 = JIIV 3 : 319.1 ' JS: 11))1 HV .

Aoo1 l}rlo1 lt z. lD)l A, f. THY ' J s:TVV 001.03. ·v l}<ho.do)l h mAJ3 A, f. nAo�l} i\01 91£'0 lw'01.091t'O lt1.o1A.?fi Alu. 3Tf l}<bndO)l U u9 3Tino1?90ltn Sn1lu.9)l11\3A. Slu. ltffl}'tff S}doox

h�t3R91tV

/i.

II •

(1 '0))1

•V

li.

' nOA(j)A. -1d1 no1 OJ3rllto <})ltd31(1}�3 tnA}3 w 01 SIJ)Ancbodlt 3191 ' no1t91 no1 <})ltd31(1}03 JHV OA(l}A}d1 01 tn)l no1t91 S<;>A3 <})ltd31(1}�3 tnA}3 w 0}3tilto n�q Av : aoMpt.zd� ao� lwroJJlJd3Jl "-�:I 1

z z» Slul\l3"(G Slu. zAll + zx z z z tll.{ + z�

0}3tilto <})ltd31(1}�3 tnA}3 W 01 3191

'1 <

< rol

·f.. , f.. 'Oi\0�1} (t01 l}1�3g 1D13)l.Q}dg (t01t ' I =� Di\}1)1'0 ll))l 0 01 Od1J\?)l 3Tf nO}"{)l(t)l -nil! no1 nJ3Tilw mAJ3 J ' g ' v n1 ' o < E z)a'M ( lD)l lD)l l}g131t3 o < (Z z)a'M ' o < (1 z)a'M ( I =d'o) n<Y'()l<pt no1 nJ3Tilw mA}3 ..1 ' g 'v n1 l}g131t3 · og31t}lt3 9)l1gnA. S3A9)113 10 ..1 ' g ' V 001.03, ' 1 = 1 E z 1 = 1 Zz 1 = 1 1 z 1 -11i 01.0 Ez 'Zz ' I z i\001

d l>AJU -n tn)l (�'n))I 01 odl-tq)l 3ti nO"()!<).)! no1 0}3tilto <})ltd3.l -(1}�3 tnA}3 w 01 3191 ' zd < z(� - t_h) + z(n - tx) * : t3<)Xot w no1 (I,\'Ix) -------

h�t3R91tV

lit JBV3 Af.ln = .

�A.gn 011.

-3rf9t1.1tC 02. l3Xdr;xua lw:fXJJ /pttdl.3rlro3tC VlOll 3:J -

hgJ3"Qlt\;j <W.1. 1Jtfl}.a:

O.L

------

-------

To B��a Tmo E�� -------

ypa<peTat : I z 2 - z3 l l z 2 l l z3 1 + I z3 - z 1 l l z3 l l z1 1 + l z1 - z 2 1 1 z1 1 1 z 2 1 ;::: l z 1 - z 2 1 1 z2 - z3 1 1 z3 - z 2 1 � I z. - Z 2 11 Z 2 - Z 3 11 Z 3 - z. I ( 1 ) . Mevez A.onrov va a1Cooeil;ovpe Trf nxet:n� (1); Arr6 o tt�T) : llpayfla'tl sxouf.1E :

l z 2 - z3 11 z 2 11 z3 I + I z3 - z 1 11 z3 11 z 1 I + I z1 - z2 11 z1 11 z 2 I >

;::: l (z 2 - z3)z 2 • z3 + (z3 - z 1 )z3 . z 1 +

(zl - Zz)Zl. zz l = lz 1 - Zz l l zz - z3 l l z 1 - z3 1

= l z; ·� -zi ·� + zi · z. -� ·� + (z. -�)z. ·�I = = l z i · ( z 1 - z 2 ) - z3 ( z � - z ; ) + ( z 1 - z 2 ) z 1 z 2 1 = = I ( zt - z2 )( zi - z3 zt - z3 z2 + zt z2 )I =

lz. -�� ( � -z.) -� ( � -z. )I = lz. -�� 11 � -�1 -z.

Ilpo<pavro<; &tav 'tO A B r eivat tcr61tA.zupo Kat 'tO K. �- 'tOU, 'tOt£ t<JXI)et ro<; t<JOTit'ta (ytati;). E t Y <p a pf.1oyt) : I:t K0.9t Tpiyrovo t.«JX'6tt �p. Axc)()tt;l) : Av to P taunmei f.16 to 1tepiKevtpo too A

P 'tO

A B r tOte a1t6 tflV 1tP011"(0Ufl6Vll CJXSCJll 1tatpvOUf.16: aj}y apy a · R2 + P · R2 + y · R2 � a · P · y ::::? R2 > =-= a+P+r 2t

4RE = 2Rp ::::? R � 2p. H CJXSCJfl a'l.l'tiJ 1tpoci1ttet Kat 2� p

a1t6 I02

R2 - 2Rp (Euler), 61tou I to tylCevtpo A B f

2!1 E <p a pf.1 oyt) : I:t K0.9t Tpiyrovo t<fl6tt. =

flaf.lp

a�

A1to()tt;l) :

A B f a1t6 A

J.lpJ.ly

�'Y

J.lyf.la � _!

Av 'tO p 'taunmei J.lf: 'tO K. �· e 'tOU

TitV 1tp0l)')'O'UJ.lf:V11 <JXS<Jl) 1taipVOUJ.1f::

a. eB. er + f3 . er. eA + y. er. eB ;;;::: a. f3 . y ' 2 2 2 dlJAaOlJ � a 3 lll V 3 lly + f3 3 llr 32 lla + 2 2 y 3 lla· 3 1113 ;;;::: a f3y 0 .......,.. lla ll(3 + llf3j3y1ly + llyyalla > 49 7COTe ....... aj3 ,

rEN I KA I:XOAIA

•

• D1tror, eivat yvrom6 11 tptyrovof.16tptldj flOp<piJ trov flt + yaotKrov apt9f.lc0v Kat 11 crxsCJll ei9 = cruv9 i11 119 oev otoacrKetat ma EUTJVtKa Ai>Keta cre avti9ecrf1 fl6 6A.a ta AuKeta it High Schools 6/..ou tOU KOCJflOU. 'Etm to ap9po . auto oev f.11topei va 9eropTJ9ei o'AoKA.TJprof.lSvo, a<Poi> Oev f.11t0p0Uf.16 va f.16AetTtCJOUf.16 "(OOVtaKSI, CJXScretr, fl6 tflV a.Ayeppa trov J.U"(aOtKcOV apt9flcOV.

• Eivat 1tepiepyo to "ff:'(Ov6r, cre 6/..o tov KOCJflO 11 uA.11 tow Ma9flf.1anKrov crtflV oeutepopa9f.1ta eK1tatOeucrf1 va au�avetat Kat crtfl xropa llar. va f.16tcOVetat cruvexror, . Eivat ouvat6v fl6 a'l.l'tiJ tflV uA.11 tfll, A.A.yeppar,, xropir, crxeoov tptyrovof.16tpia Kat EudeiOeta rerof.16tpia, va fl1tOpscret tvar, <POttfltfll, va 1tapaKoA.oulh'Jcret Kat 1toA.U 1t6ptcrcr6tepo va a<POflOtrocret ta Avrotepa Ma9flflattKa; • DA-11 11 EuKAriOeta ferof.16tpia f.11tOpei va �9ei fl6 tfl 13of19eta tfll, aA.yeppar, trov flt"(aOtKrov apt9f.lc0v , 61tror, J.UtCr pei Kave{r, va ommmrocret mo [ 1 ] tfll, PtPA.tO"fpQ<Piar,. H f.16AS'tfl al>tfl eivat aA'Aote 6UKOAOt6P11 , a1t6 6tt 6tav yivetat f.16 tflV KA.acrmldj 1tapayroytldt fl£9ooo Kat O.Uote avayetat cre 1tOAi>1tAOK61, aA.yepptKSI, 1tp�etr, . :Ee Ka9e 1t6pi1t'troCJll 11 u1tep0Xli trov f.169oorov tfll, EudeiOemr, rerof.16tpiar, cre OflOP<Pta eivat KataA.unldj .

B I BA I OrPAcJ)IA

1. T. Andreescu , D. Andrica , Complex Numbers from A to . . . Z, Birkhauser, Boston , USA , 2006 . 2. R. Deaux, Introduction to the geometry of Com plex numbers, Dover Publications, Inc., New York,

2008 .

3. A. Engel, Problem-Solving Strategies, Springel, New York, 1 998. 4. L.S.Hahn , Complex Numbers and Geometry , M.A.A, USA, 1 994. 5. P.S Modenov, Problems in Geometry, Mir Pub lishers, Moscow, 1 979 .

6. M.r MapuyKcitcrt�. l:Totxeiu MtyuotKoov Apt9J.lrov, T.E.I Hp mcA.eiou Kpft'tlJ�, 1 995. 7. H. Schwerdtfeger , Geometry of Complex Num bers , Dover Publications, lnc, New York, 1 979 .

• I:xoA.to TTJ� I:l)vTuKnKft� ErrtTporrft�: llpo<pu voo� TJ 1.:6CJT) crTO rrp6�ATJJ.l.U 4 (®SJ.l.U E.M.TI. 1 959) sivat Um>"(KpiTro� euKOAOTepTJ uno TT) AUCJT) 1tOl) 069qKe TOTe f.1E �acrl) 'tl)V TptyrovoJ.leTpttct1 J.!Op<pft z = p ( ou�9 + iTJ !l9 ) Kat TOV rorro De Moivre. 'ETcrt

oturrtcrToo9qKe on Tu CJT)!lf:iu A,B,r �piO"KoVTat os �ta 'tOU cl�OVU y 'y, U<j)O'll

( )

Re 1 - zv = 1 - pvcruv ( v9 ) > 0 yta 0

= 1, 2 , 3 . Erro,...S-

vro� TO eivat eKT6� Too Tptyoovol) ABr. EtotKa uv z � IR, OlJAaOTJ z = x + yi flB y 0 , TO'te s�ucr<paA.�eTat To on Tu A,B,r eivm KOpl)<pe� Tpt yoovol), u<pou

(

)

det AB' Ar =

(

... = y x 2 + i

EYKAEIAHE B' 89 T.l/68

i - x2 + x

y ( 1 - 2x )

(

3xi - x3 + x Y i - 3x 2 + 1

)[(

x - 1 )2 + i

]*o

)

� o �

E u KA&i onc:

n p o-.:&iv&l . . .

Enaptuw: r. K. TPIANTOI: -

( TEYXOTI: 83 ) Av k , t6't& va A.ua&i TJ e�ioroOTJ x4 +(1-2k)x3 + (k2 +2)x2 - (k2 +4k -5)x+3(k-1)2 =0 ( fl.QPrm: AITOI:TOAOITOY i\OI: - MeooA.6yyt) AYI:H ( riQPfOI: �Ei\HI:TA8HI: - ASijva ) EKt&A.ouf.l & t� 7tp�e� otataooovt� 'tTJV e�ioroOTJ ro� 1tpo� 'tTJV 1tapaf.J.&tpo k,Kattxouf.l &: {x2 -x +3)k2 -2{x3 +2x +3)k+x4 +x3 +2x2 +5x +3 = En&!Oit x3 + 2x + 3 = (x + 1)(x2 -x + 3) Kat x4 + x3 + 2x2 +5x +3 = (x + 1)2{x2 -x +3) ( 1) � {x2 -x+3)[k2 -2k(x+1)+(x+1)2 ] =0 1±iv'il � (x2 -x+3)(x+1-k)2 =0� x = 1 - k v x = --2 Auoe� totetA.av &mOTJ� ot <rovaoeA.qxn: Po06A.<po<; M7t6pfJ<; - �a<pVfJ, fuopyo<; Mittoto<; - Paf.J.ta Apt�, �lOvU<JfJ<; ruiwapo<; - llupyo�. ruiWTJ<; HA.t67touA.o<; KaA.af.J.ata. ( TEYXOTI: 83 ) �ivetat tpiyrovo ABr f.1& AB , o eyyeypaflfl£v� e<pametat trov AB, ma Z, E K'6� {l, p) avriototxa, o napeyyeypaf.1f.l£v� K'6� {la , p ) &<pam&tat trov AB, ota Z1 , E1 avrimmxa. H tEf.lV&t ZE <JtO L Kat ZIEI <JtO :El . Na OetxEl&i I:I:1 I / {OPEI:THI: KATIANOI: -llptj3f4a) PO�Oi\OI: MllOPHI: - Mt<pvfJ ) = Eivat AE AZ = t-a, AE1 = AZ1 = t OTJA. ta tpiyrova AEZ Kat AE1Z. eivat too<JK&A.it f.l& 1t - A , 01t0t& ZE I IZ1E1 it I:E I /I:1E1 (1) AEZ = AE1Z. = -2 1t-A , I:to tpiyrovo fLE Ex,ouf.l&: :ErE = -2 ,:LEA = -2 1t-A r Ef = t -yKat EI:r = I:EA-I:rE = -- -2 = -B2 . 2 I:uve1tro�. a1t6 tov v6f.1o trov llf.1tt6vrov 1raipvouf.1&: r r I:E Er -= r -B => I: =Er -B2 ={t-y) -B2 (2) 2 2 2 2 I:to tpiyrovo rLI El ExOUf.l&: rEI = 't-p Kat 1t-r- - -·1t-A A+r = -1t-B &VcO a1t0 'tOV -fL1 E1 =1t- = 2 <Jto2 tpiyrovo2 am6: 2 v6f.1o -rrov T)f.1tt6vrov . -2- => I:IEI = rEI => I:1 1 - rEI · TJf.l-1t-r 1t-r 1r-B 1t-B -2 2 -2 AI:KHI:H

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re�; la!:"1 Z1 = rE1" E1 = 1t-!:1 rE" 1 -!:1 E" 1 r = 1t- 2 1t-A 1t- -A+B ---=---1t-A 21t-A-B-1t+A ---= 2 2 2 2 " " " B e�; 1t-B = -2 =-=1 2 aBZ1 => Ia!:1 Z1 =Ia BZ1 => To la!:1 BZ1 eivat eyypa'lftJ.l.O ae o1t6'te la!:"1 B = 1t-la Z"1 B = 1t - l1t = 21t => lar .lB!:1 => lar .l BT1 . !:'to 'tpiyrovo rBT1 11 r!:1 eivat �tX.O'tOJ.l.O� Kat U'lfO� Kat apa �tUJ.l.tG�, 01tO'tt !:1 BT1 1C6KA.o,

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E<papJ.1.6�ovtac; crraupoet.Oit 1tOAicnaaJ.1.6 ouo npclne<; �taroae� 'tou aucrtitJ.l.a'to� (E') , txouJ.l.t: x-a = y-p = z-y =k <:::> lr� :PI 1:P P�� lP� r�l x-a y-P . z-y a(p-y) = p(y-a) = y(a-p) =k <:::> {x -a= ka(p-y) {x = a[l +k(p-y)] y-P =kP(r - a) <=> y=P[l+k(y-a)] (T) z-y =ky(a-p) z=y[l +k(a-p)] Me avnKa'tacrraan 'tow x,y,z a1t6 axta� <n, antv 'ttA.etrtaia e�iaroan 'tou (!:') , 1taipvouJ.l.t: a2 [l +k(p-y)]2 + P2 [l +k(y-a)f + +y2 [1+k(a-p)]2 = a2 + P2 + '/ <:::> a2 + P2 +y2 +2k[a2 (P -y) + P2 (y -a) +y2(a-p)] + k2 [a2 (P- ri + P2 (y- a)2 +y2 (a -p)2 ] = a2 + P2 +r2 <=> k2 [a2 (P- ri + P2 (y-a)2 +y2 (a-p)2 ] + 2k[a\p-y)+ P2 (y-a)+y2 (a -p)] = 0 <:::> k=O (A) a) k = a2(p-y)22(a+-p)(p-y)(y2P (y -a)2 +y2 (a-p)2 fta k=O , a1t6 't� CJXEGt� (T) , 1taipVOUJ.l.t ntV 1tpO<pavft Man: x = a,y= P ,z=y . na k a2 (P- ri2(a-p)(p-y)(y-a) + P2 (r -ai + r2 (a -Pi a1t6 axtae� (T) 1tpolC61t'ttt 11 �ro'ttP11 A.uan 'tOU (E). ( Ta 1tapa1tavro t<JX6ouv 6'tav a# p # y # a) Av a = p = y # , 'to apx.tK6 aU<JntJ.l.a ypa<pe'tat: x+y+z=3a { x+y+z=3a x + y + z = 3a <:::> x2 +y2 +z - 3a2 (P) x2 +y2 +z2 =3a2 A6yro 'tO>V axtaerov (P), 11 'taU'tOnt'ta: (x + y+ z)2 = x2 + y2 + z2 + 2(xy+ yz+zx) �ivet 9a2 = 3a2 +2(xy+yz+zx) <:::> xy+yz+zx = 3a2 �11A.. x 2 + y2 + z2 = xy + yz + Kat 'teAtKa: (x -yi +(y-z)2 + (z-x)2 = 0 <:::> x = y = z o1t6'te: x = y= z = a . Auae� tcrraetA.av, eman� �tA11G't ot auvooeA<pot: a911� - A9itva. TEYXOYL 83) ( �ivt'tat 'tpiy'trovo crro Ant� Ar AB Kat0 xepiJ.l.t po ianABrJ.l.top9oyrovto 'to u1t6A.omo �wipean� 22 22 :13 . !:'to 11J.l.t.e1tixe�o 1t0\) op�et 11 Br Kat oev avftKtt 'tO A, aeropOUJ.l.t GrJJ.l.tia E rocrre: B�//Ar,rE//AB,Bil =rE < AB. Av Z 11 'tOJ.l.it 'tO>V r Kat BE Kat H GrJJ.l.tlO 'tll� 11J.l.teU9eiac; 'tt'tOtO, • • •

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(AH) = J2 · p . ll>epouJ.W HA AB, HM Ar Kat e7tetOftKat-ro-roH 'tavftKet Q"tTTV AAHM OtXO'tOJ.l.Oeivat -rtaxl>et 6n HA=HM e tpa1tAI>Upo er payrovo va J.W MO'tBAeOJ.l.a eivat (AH) = (AA).fi. = p.fi � (AA) = p . .i\pa (AA)= (AM)= (MH) = (HA)= p . �UVt1tcO�, 'tO OTJJ.WtO ABf KatV -roapaJ.l.ft 'tll� E>H J.W H eivat ByKSV'tpO 'tOOvoJ!at; U 'tptyrovou Tou ABr. o uJ.W K 'tl'I 'tl'IBE>K V AB.Kat HapaBHOtaJ.Woo�. eivat OtXO't�'toOJ.l.op6oyrovto O� Kat U'lfO-rp�a1tB�to cno -rpAKE>I iyrovo HM eivat OtaJ.WoO�, 01tO'tt 'tO M eivat 'tO J!Boov -rou. Al. E1toJ!Bvro�, taxl>et: (AI)=2(MA)� 2=2p�p= l E1tttOft, OJ.l.OO� eivat (AM)=(M)='t-a=:>1=6-a=:>a=5. A1t6� oxeoe�: P +r =2-r-a=12-5 = 7 Kat �2 +y2 = a2 = 25 , Myro. y < � , 1taipvouJ.W -reA.tKa 6n: �=4,y =3. Tpe� <piAOt eixav KU1t0ta�uJ.l.XPOVtKft CJ'tt'YKa6eva� J!it MO a1t6 eva XP'll J.l. a nK6 1too6v o Ka6ev�. Ka6ev� A6ijva aPXft. Na �pe6ei n 1tOOOV ot86t't ) 'tOU� CJ'tl'IV E>eropOUJ.W 'tOU� KUKAOU� c; : (O,R) Kat (O,p) R p Kat OTJJ.WlO A 'tOU KUKAOU . Tuxaia llJ.1teu6eia a1t6 -ro OTJJ.Wio A 'tBJ.l.Vtt yta 1tPcO't'11 <popa -rov o-ro OTJJ.Wio B. A1t6 -ro <pepouJ.W llJ!teuOeia K'61CAo BfLl 1tOU 'tBJ.l.VI>t 'tOV cno r Kat 'tOV cno Ll B'tot, rocne va eivat ABLl = 72° . E>eropOUJ.W, OOOTJ�, xopoft BE 'tOU Ka'ta 'tp01tOV rocne Ot yrovie� ABLl Kat LlBE va eivat e<pe�ij� Kat va taxl>et a:fm = 54°. (.i\pa � ABE = 126° ). 'Ecnro eAA&t'lfll :-+-= a2 p2 1 J!t a= (AB), � = (Bll) . Av -ro tJ!�aoov -rou xropiot> 1tou 1ttptKA&iet eAAtt'l'll au'tij eivat ioo J.W -ro tJ.1Pa06v 'tou KUKAtKOO OaKmA.iou 1tOU opi�ouv Ot KUKAot Kat Kat (BfE) = 2-Jil>+ 2, 61tou o Myo� 'tll� XPUoiJ� 'tOJ.l.it� 'tOt€ va U1tOAoyto6ei aKriva p 'tOU ltMov 'tO tJ.1Pa06v 'tou OaKWA.iou eivat ioo J.W 81t(3-ll>) , E) . -r&te va unoAo-yto6ei o Myo� -rrov �>J!Paorov: (AB (fBE) PaJ!ta .1\p-ra�. A<pteproJ!Bvo o'tll (J'I)'Eppou, Vooi>A<po a1t6 'tl'I V OpeO'ttaOa 1toua1toOetxeei1toA.u vrop� .....-r)piyrovo ABf J.W J.l.iJK'll Na 6n oe Ka6e 1tAI>Uprov a, �, y Kat aKrive�avno't R, p -roroixro�, v 1ttpttaxl>tt ou , yeypaJ.1J!Bvol1tAft enerpaJ!J!Bvou KUKAroV '1 1 , J3 1 avtOO'tll'ta: 2R -a+ P + --+ y+a <-2p . � +y -MeooMyyt ). l_

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. . . H aJ..tj8e1a eivw orl i:xovp.e 1!apa 1rolli� KW ac;loA.oye� epyauie� alla Kw epo:n:f!p.ara. Fla oae� i:xovp.e Kal!ow al!aVU/UI'f rrf Jytp.omeiJovp.e. Era Gl!op.eva re6xrt Ba vl!apxel uvvi:xew Kw yw rl� ille� epyaaie� l!OV 1rolli� al! ' avri:� Oa t5'!}lOmev8ovv. Evxaplmovp.e yw rrfV KaTaVO'If5'1 U.l� mfJavi:� Ka8vmepflael� UrrfV Bl!IKOIVWVfa B 1 py 1v i a 'l'uxov. 01 owmmrocn:� crac; cmJV eucacria Go1dbach eivat crrom£c;, af..J...iJ. o 1tpOTJYOUJ.lEVoc; Kill o E1tOJ.lEVOc; Kaee 1tpcO'tOU, OEV eivat 1tUV'ta ap-rtoc;, yw 'tOV JlOVO Up'ttO 1tproto 1tOU yvropfSOUJlf: mlJlf:PU 'tOV 2 1tptv icat Jlf:'tU OEV eivat apnot fi).)..a 1tEptnoi. Kwai Chang Caine. XatpoJ.ta<ne 1t0u f:xete Kat ota�a�e'te aKoJla 1taA.ta tet>xrl TOU 1tep1o01KOU J.tac;. H 1tapatiJpTJcriJ crac; cmJV AU<rrJ 'tl]c; e�icrro<rrJc; Vx + 20 - Vx + 9 = . 1 1tou OTJJ.!OmeU'tTJKE <no tf:Uxoc; 38 eiva1 crrocrtiJ Kat apKETa evotacp£poucra. E>a 'tTJ OTJJ.!OmeUcrouJ.!f: <no E1tOJ.lEVO Twxoc;. 1\.yyd.o� Kp aTT] JlEVO�. H AU<rrJ 'tl]c; 'tpt'tO�aeJ.ttac; e�icrro<rrJ c; roe; etJ.ta txet eoro Kat XPOVW f:�UV'tAT]eei. �EV eivat cracptc; To epro'tl]J.ta crac; 1tepi v£ac; J.!f:eooou Jlf: e�a.Aet'I'TJ Tou oeuTepo�aeJ.ttou 1tapayoV'ta. M1topehe va !lac; crTeiAETe Ka1toto crxenKo 1tapaoetyJ.ta; H paKA.l) � ll a1taato1tovA.o�. �EV �£pouJ.tE av U1tUPXOUV e�royl)tva oV'ta. H J.ta8TJJ.tanri) crac; a1tooe�TJ yta 'tl]V U1tap�TJ e�royl)tvrov ovTrov etA.et otopero<rrJ. To ytvoJ.tEVO ouo apteJ.trov 1tou o £vac; Teivet cr'to JlTJOEv Kat o a.Moc; <no a1tetpo oev eivat 1tUV'ta a1te1po. A erovio a� ITa1tavTroviov [reronovoc;]. H aucmJpl) J.tU8TJJ.tUnriJ A.oytKl) U1tUPXEl <nouc; optcrJ.touc; 'tl]c; fEroJlf:Tpiac;. 'Icrroc;, OEV f:xete EtK6va, on <na crxoA.eia 1tAEOV, OEV KUVOUJ.!f: 'tU 1tUV'ta U1t0 'tl]V �troJ.tanKl) eeJ.!f:A.iroOTJ. �EV a1tOOEtKWOUJlf: 1tAEOV 6n TO Eva eivat Jlf:YaAUTEpo a1to TO Jll]OEv. fta 'tl]V U1tap�T] 'troV KUVOVlKOOV 1t0A.uyrovrov (icrec; 1tA.eup£c; icr� yroviec;) f:xouv f:t1t(J)eei 1toUO. vrophepa Kat oev eivat Ti1rora au8aipeto. �e 1tnAta �t�A.ia, 'tl]c; of:Kaetiac; TOU 70, 1t0U eixav OtacpopenKl) eeJ.!f:A.iro<rrJ 9a �pEi'tf: 'tE'tOtf:c; U1to0f:�E�. fta 'tOV optOJ.!O 'tl]c; tOO'tT]Tac; eu8uypaJlll(l)V 'tJll]JlU'troV 1tOU roe; OT]Jlf:tocrUVOAa OtacpopenKOOV <rrJ Jlf:lroV 'tOU xropou, t<JXUp\SE<nf: 6n, OEV Jl1tOpei va eival icra 1tapa J.!OVO 1cro&UvaJ.ta Jl1tOpro va crac; 1tro Ta e�l)c;: �'tTJV 1tpayJ.!UnK6'tT]Ta oev f:xouJ.tE icra axr'lJ.taTa acpoo oev cruJl1t{xTouv Kat Eivat owcpopenKa OT]Jlf:la 'tOU xropou. DJ.!(l)c; opfSOUJlf: roe; icra O'tUV Jl1tOpoUV Jlf: KU'taAAT]AT] Jlf:'tUcpopa va cruJ.t1tEcrouv. �uo Tpiyrova 1t.x. AtJ.tE 6n eivat icra 6-rav TO tva Jl1tOpei va JlETacpepeei 1tcivro <no a.Mo Kat va auJ.t1tEcrouv. �'tTJV 1tEpi1t'trocrl] autiJ OAa 'tU <notxeia Touc; eivat icra Kat 'tU EJl�Uoa Touc;. DJ.tOOc; f:xouJ.tE Kat 'tpiyrova 1tOU evro OEV eivat icra f:xouv icra EJ.t�aoa Kat Ta AEJlf: tcro&UvaJ.tU. �Ta euBUypaJ.tJ.!U TJltlJlUTa E1toJ.!Evroc; f:xouJ.tE JlOVO 1cr6'tl]Ta o:xt tcroouvaJ.tia. �uo euBUypaJ.!J.!U TJltlJlUTa Jlf: Ka'taAATJATJ JlETacpopa Jl1topoov va cruJ.t1tEcrouv Kat TOTE ea f:xouv Kat To iOto JltlKoc;, £-rm Ta AEJlf: icra Q:xt tcro&UvaJ.ta o1troc; Ta Tpiyrova 1tOU f:xouv JlOVo icra e�a. NiKO� X a'r �l) � [13 xvovrov, JLUOqn]c;] . Kaee �uyoc; (ap-rtoc;) apteJ.toc; Jl1tOpei va ypacpei roe; aepotcrJ.ta ouo 1tproTrov apteJ.troV Met TJ etKacria Tou Go1dbach. Me 'tl]V epyacria crou ecrU a1t£oe�ec; on oev t<JXI}et, oJ.troc; auTo oev eivat crromo. Aec; yta 1tapaoetyJ.ta on «oTav £vac; apteJ.toc; X txet K 'I'TJcpia o 2X ea f:xet K+ b. AuTo oJ.troc; oev eivat crro<no, OEV f:xet 1tclV'ta K+ 1 'I'T]cpia, Jl1tOpei Kal auToc; va f:xet K \fll]cpia. EcppoauvlJ 1:Ta!JaTo1touA.ou. �'tTJV epro'tl]cri] crac; nota eivat 'ta KpttiJpw otatpeTO'tl]Tac; evoc; apteJ.to6 JlE TO 1 1 Jl1tOpOUJlf: va crac; 1tOUJlf: Jlf:ptKa. 1 ) MtKpo eeropl]J.!U 'tOU Fermat. ApteJ.toi o.P-a 01tOU a cpumKoc; Kat p=l l . 2) ApteJ.toi 1tOU 'tO 1tpcO'tO, 'tpi'to, 1tEJl1t'tO, KA7t 'I'T]cpia f:xouv 'tO tOtO aepotcrJ.ta Jlf: OeU'tEpo, 'tE'tllp'tO, EK'tO, KA7t 'I'T]cpia ei'te TO 1tA.l)eoc; Touc; eivat �uyoc; ei'te eivat J.tOvoc;, 1t.x. 651343 1) 143 1tproTo Kat Tpho ocro TO of:UTepo. Y1t01tEpt1ttrocretc; Tou 200 Kpt'tl]piou eivat: i. Apt9J.toi o1troc; o 1 2 1 , 12345432 1 , au�avouv Ta \fiTJcpia Tou KaTa tva Kat aKoA.oueroc; J.tEtrovoV'tat KaTa tva. ii. 1\p-rto apteJ.to iOtrov \fiTJcpirov Kat <no TtAoc; 1 1 , 1t. X· 77 1 1 , 2222 1 1 , 9999 1 1 . iii. �'tl]V aPXJ1 Kat <nO 'tEAoc; 1 1 Kat cmJ JlEOT] Jll]OEVtKU, 1t.X. 1 1 0 1 1 , 1 1 00 1 1 , 1 1 000 1 1 . iv. �'tl]V aPXJ1 Kat TO 'tEAoc; 0 iOtoc; 2\f1tlcptoc; apteJ.toc; Kat cmJ JlEOT] JlOVoc; apteJ.!Qc; Jll]OEVtKOOV, 1t.X. 53053, 1 20012. v. L'tTJV aPXJ1 Kat TO T£A.oc; o iotoc; v-'f1tlcptoc; apteJ.toc; apKei va otatpei'tat JlE 1 1 Kat O'tTJ JlEOTJ JlOV6c; apleJ.toc; Jll]OEVtKrov, 1t.X. 1 2 1 0 1 2 1 , 1 3 3 1 0 1 33 1 , K.a. E J.t!JilVOUl)A. Mavo�. To epcO'tT]Jlcl 1tOU t<JXUpfSe<ne eivat on: «av txe� 32 xap-raKta 1tOU ava 4 txouv 'tOV ioto apteJlo a1to 0 £roe; 7, oTav otaAt�e� wxaia 4 n 1tteavo'tl]Ta txouJ.!f: TO clepotcrJ.ta Touc; va eivat 0,_1 ,2,3, . . . ,28;». To 1tpO�AT]Jlcl AoUtOV eivat cnJV&t>O.GJLOi JLE E1tO.VclA11'1'11, EXE� 4 9£cre� Kat e£Af:� Va �aAE� cre Kaee 9£0T] £vav a1t0 'touc; apteJ.touc; 0-7 E'tm 'tO Jll]OEv roe; aepotcrJ.ta JlOVO JltO. cpopa Jl1tOpei va 1tpoKU'f1Et <no cruvouacrJ.tO 0,0,0,0. To tva Jl1tOpei va 1tpOKU'f1Et 4 cpop£c; <nouc; cruvouacrJ.touc; { 1 ,0,0,0 1) 0, 1 ,0,0 1) 0,0, 1 ,0 1) 0,0,0, 1 } , . . . , TO 28 JlOVO JlE TO cruvOU(lOJlo 7,7,7,7. �e Kaee 1tepi1ttrocrl] OTJAnOl) ea �pe� a1to 1tOtouc; apleJ.touc; Jl1tOpei va txe� MpotcrJ.ta TO croyKEKptJ.!Evo apteJ.to ( . . . ) K.A..1t .. E�eT�e� OTJAMtl oA.ec; � eKooxtc;. Al]JltlTPTI � Ka p peA.a � [VIJLlKOc;] . Euxapt<nOUJlf: yta Ta Tpia eerop1)J.ta'ta fJE � a1tooe�e� Touc;, cmJ eeropia apteJlffiv Kat f:xouv va KUVOUV fJ£ 'tT] OtatpetO'tl]'tll Kat eivat <JXE'ttKU Jlf: 'tO eeropl]Jlll Fermat.Acpou Kpteoov on eival <nouc; crK01touc; 'tOU 1tEplOOtKoU ea OTJJ.!OmEUeouv. ·

EYKAEIAID; B'

89 't.l/72

Eod� ypafP&Tt

-------

• • •

tpd� anavtapt

--------

Ao p tvT�o na�oTTt. Av mcrte6eTe, 6tt f:xtte �pet K6.tt Katvoupyto, ot owave� enbttoo, npenet va to crttf.A.ett OTtJ.I.OoleuCJll ot K6.now t7ttCJTTt J.I.OVtJC6 7ttptoOt1C6, tt'toto eivat Kat TO MaeTtJ.I.aTtK6 �eA.tio TTt� EME, nou elCOWt'tat CJTTtV AyyA.trij yA.roooa. KroGTa� Mav11� [M«OllfJ«nKo;, AOqva) An6 tov 1CaA.6 q>it..o Kat ouv6.oel..q>o niJpaJ.I.e CJXettrij tmotoA.t'J yta TTtV q>rotoypaq>ia tou APXtJ.I.TJOTt crto 7teptOOt1C6 J.l.� [TeUXO� 87], M6.ptto� 20 1 3 ot CJXett1C6 apepo: «cl!CA.ou J.I.STPTtCJ�». H tpffiTTtOTI tou tivat, av. auTt'J tivat np6.yJ.1a'tt 11 aA.11etvt'J, ytati npooOJ.I.Ot6./;tt J.1t TTtV q>rotoypaq>ia tou �amf..t6. TTt� I:x6.p'tll� APXiOaJ.I.OU [I:xoA.t1C6 �t�A.io A' fUJ.I.Vaoiou otA.iOa 84]. yta

Ot OtiC� J.l.� avaq>op� tivat ot apKeTt� q>rotoypaq>ie� an6 ap1CeT6. J.l.aflll J.I.aTtK6. �t�A.ia tou 1 6<10 - l7<1U atrova, 6xro� Kat ottpe� ypaJ.I.J.I.atoot'JJ.I.ffiv 1tOU f:xouv elCOoeei ax6 Otclq>Ope� xrope�. EvoetlCTtlCcl 011J.I.Oote6ouJ.I.t J.1tpt1Ct� ax6 aute�. Aq>t'JvouJ.I.t 6J.1ro� avotKTO to eeJ.I.a, yta xoUou� ouvaott..q>ou�, etaorott� au'tt'J � TTt� totoptrij� ava/;t'JTTtOTI� Kat 1teptJ.1SyOUJ.I.t av'ticrtOtXIl CJXettrij 1tA11POq>OP11CJll , 1tOU J.1t xap6. ea 011J.I.Oote6oouJ.I.t.

I:xoA.to: 11« -ro •

E1tetoit 0.1t0 opta�ouc; auva.OEAq>ouc; J.lOU Ota.nmro9r}Ke T) a.1topia. yta.ti va. J.lT) oextoUJ.le 'tT)V tOO'tT):tO. V = V . A * mate va. J.l1tOpOUJ.le E'tat va. opiaOUJ.le Kat OUVUJ.letc; VK J.l8 K=3,4,5 ... , exro va. e1ttOT)J.lUvOO

'tO. el;itc;:

1t.x. op�«JlE

V = V2 v(= V . V2 ) ' 'tOte v 3

•

E V Kat J.laAtata.

v it V ' xropic; OJ.lOO<; 3

Ka.eopta�o aUVOAO optOJ.lOU 'tT)<; 1tpU/;T)<;. E>a. exet apa.ye aUVOAo optOJ.lOU 'tO JR+xV it to VxlR + ) ; ) . Me V 1ta.ptmavouJ.le to a'6voA.o trov et..eU eeprov ota.VUaJ.latrov tou em1teoou it tou

(

XOOPOU. E/;UAA.ou «V op{�«JlE

V-4 = V-3 · V- ( = -V · V-3 ) it V-4 = V- 2 · V- 2

(

ea. eiXO.J.le

V-4 E JR + , a.Uii roe; aUVOAo

optOJ.lOU 1t0t0 ea. eixa.J.�.e(;). To VxV it 'tO JR +xlR+ ; ) . Eoro eiva.t eJ.lq>O.VEatepo O'tt oev EXOUJ.le

Ka.eopta�o aUVOAo optOJ.lO'U 'tT)<; 1tpU/;T)<; a.Ua Kat O'tt 01tOtOOit1tote eK 'tOOV ouo eeropitaouJ.le ea. eiva.t Ota.q>opettKO 0.1t0 'tO. 1tpOT)yOUJ.levO. JR+xV ' VxlR + . Ot 1tpal;etc; Aot1tOV tOtO aUVOAo OplOJ.lOU.

V-3 'V- 4 0.1tOKAele'tat va. exouv 'tO

AvaA.oya. 1ta.paool;a. ea. eixa.J.�.e yta. ta. 5 , 6 JCI.Jt. E>a. Ka.ta.Aitra.J.le A.ot1tov mo 1ta.paA.oyo au J!1tEpa.aJ.la., ot 1teptnec; OUVUJ!et<; va. 1ta.ptmavouv OtO.vUOJ.lO.'ta. evro ot apnec; a.pteJ.lOU<; Kat J.laAtata. xrop� f:Vl«io mlVOAo OplGJlOV.

•

EI;UUou, ytVOJ.levO 'tT)<; J.lOpq>itc;

a.�.r

it V . V .

v( = V ) ' OeV exet VOT)J.lO., a.q>ou, 01t00<; eixa.J.�.e ypa\jlet

ato 'TEVXO� 8 7, 1tpOU1toeetet 'tT)V 1ta.pliA.oyt) tOO'tT)'tO.

(

(a · � ) · y = a · ( � · y) . �'tT)V tOO'tT)'tO. a.uti} J.l8 'tO

ioto aUJ.l�OAo . ) evvoouvtat (Ka.Kro<;) tpetc; Ota.q>opettKE<; 1tpal;etc;, 0.1t0 'tl<; 01toiec; J.laAtata. T) J.llO. oev •

exet KO.V optatei, xropic; q>UatKU va. yiveta.t auto O.vttAT)1t't0 eK 1tpcO'tT)<; 0\jleroc;. Auto Kat J.lOVO ea. J.l1tOpouae va. 0.1tOtei..Ecret 0.1tUV'tT)OT) O'tT)V 1ta.pa.1tUVOO 0.1tOpia.. Me ta. 1ta.Aa.t6tepa. axoAtKa �t�Aia., �e�ata., crta. o1toia. u1tftPXe E>eropia. OJ!aorov Kat �ta.VUaJ.la.ttKrov xroprov, itta.v euKOAOtepo va. a.1toq>euxeouv 'tE'tOtOU eioouc; 1ta.pa.voitaetc;. ruopyo� �. T«GG01t01>Ao�

* BA.txt Eu!CA.eio11 B', te6xo� 87, oeA.iot� [46-47] 6xou emxttpftflllKt va yivtt Katavo11t6 6tt 11 np6.91

A · v sxtt

cr6voA.o optOJ.I.OU 'tO JRxV ' evro 11 V . A oev f:xet optcrtei. AU6. Kat av eixe optcrtei, ea eixe cr6voA.o optOJ.I.OU 'tO VxlR Kat 6xt to lRxV . E>a exp6Kttto A.om6v yta ouo otaq>opettKt� xp�e� EYKAEIAm:. B ' 89 -r. lf/73

�L/J"l.

6 8 , 8 :IHVI'JlV)IA'Jl

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