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IIoA.A.oi 'EllTJVS� TIJV sxoxiJ SKStVTJ �sKt­ vouv yta TIJV Eupro1t11 , 'tTJ Procria Kat aA.A.ou yta va spyacr'touv Kat va crxouoacrouv.

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Eq>TJ�O� a<pTJO"S 'tT)V 1ta'tpioa 'tOU 'tTJ Xio Kat tcp9acrs O"'tTJV l'taA.ia, 01tOU cruva­ V'tT)O"S 'EllTJVS� xou 'tOv �oi]9TJcrav. TsA.tKa t­ f..LStvS O"'tTJ Bsvs'tia, f..LBA.o� TIJ� EllTJvtldJ� KOt­ VO'tTJ't� Kat 'tOU «<ppoV'ttcr'tT)piou» TIJ�.

(1590-1596)

(Mavoui]A.) r11n;oovq�

BIBAION

OPOllioiPON fOI.S TIA!J, n&rtt•o• TWH ra. nrAI'fiARif APJ.J4K1'110ltt ,_k4Af1H .. llt.l'.

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EKsi, svro crxouoass, iopucrs SKOO'ttKO OtKO Kat acrxoA.ou'taV f..lS 'tt� Otop9rocrs� Kat 't� s­ K'tU1tcOO"St� �t�A.irov. 0 ioto� typmvs <<BzfiA.iov npoxezpov roz<; no.(Jl nepztxov rcvv nparcrzKdJv

!:'to �t�A.io amo, 0 fAUsOUVTJ� 1tSptAaf..L�avs 1tpO�Af]f..La'ta xpaKnld)� apt9f..LTJ'ttld)� Kat av'tA.oucrs ta xapa­ osiyf..Lata 'tOU a1t6 'tt� Ka9TJf..LSptVE� Sf..L1tOptKE� cruvallayt�.

H xproTIJ EKOOO"TJ 'tOU tytvs 'tO To �t�A.io tytvs yvrocrt6 Kat ro� "fA.utsouvt" Kat yvroptcrs f..LS'Y<iATJ sKoonld) sxtroxia acpou E-yt­ vav 21 yYOOO"'tE� S1taVSKOOO"St� 'tOU EOO� Kat 'tO Ka9ro� 'to XPTJcrtf..Loxostoucrav crav crxoA.t­ Ko �t�A.io O"'tTJV xpocrxa9sta otoacrKaA.ia� Kat otaoocrTJ� TIJ� sxtcrTIJf..LovtldJ� yvrocrTJ� xpoKst­ f..L€vou va A.u'tprocrouv to ux6oouA.o t9vo� ax6 'tT)V ayvma Kat 'tt� 1tpoAi]\j1St�. H ypacpi] 'tOU si­ Vat O"'tTJV sdaiKsuf..l€vl'J yA.rocrcra.

TTJV xp6cr9scrTJ 'tTJ ypacpst a6vo11f1'/, TIJV a­ cpaipsO"TJ V'fJ1'/Af.J.OV, ta dacrf..Lata nmKiap.am, 'tT)V a1tAo1tOlTJO"TJ UJ.lap.ov. o fA.utso'6VTJ� acpTJcrs crTIJ ota9iJKTJ 'tou tva f..LS'YaAo xocr6 oouKata) yta va K'ttcr'tsi crxoA.sio O"'tTJ Xio Kat yta 'tO f..Ltcr96 'tOU oacrKa­ A.ou xou 9a oioacrKs crs auto svro yta va f..LStVst tou <i<pTJcrS to xatptK6 'tOU crxin. Ms TIJV KATJ­ povof..Lta aurit i]9sA.s va os�st x6cro O"TJf..Lavnld) f]'taV TJ uxap�TJ • <J"XOAstOU yta 'tT) f..LOp<pOOO"TJ 'tOOV sllf]vrov 1tOU � taV TIJV S1toxfl SKS{� U1tOOOU­ AOt O"'tOU� 'tOUpKOU�. apz()p.1'/ilKchv � i1'/V A.oyapzamzK�V».

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0 ME9o(ho<; Av9paKiTI)<; (1660-1763) a1t6 'ta Zayopoxropta, 61tou y�9TJKE, yp6.­ q>TJKE <J'ttl fKtO'UJ!SlO :I:xo/..� JlS oteu9uv� 'tOV rEropyw l:o'UyOo'Up'l\ 1tOU oioacrKE <l>t/..ocroq>ia Kat «Ns6'tEPTJ <l>umKf]» a1t6 'tO PtPA.io 'tou. 0 Av9paKi'ttl<; cr1to6oacrs cr'ttlv l'tal..ia, Jl<l9TJJ.tan­ K6., Acr'tpOVOJ.tia Kat <l>t'Aocroq>ia. Emcr'tptq>o­ V'ta<; <J'ttlV EAA.6.oa, y(>pro (J't(l 1708, oioa�E (J't(l ftaweva Kat <J'ttl Kacr'topui 6nov avtt..aps Tl1 otz69UV<JTJ 'ttl<; avffi'tepTJ<; 't01ttKf]<; <JXOA�<;, 'tO

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A1t6 't(l PtPA.ia 'tOU otacrffiSTJKe 'tO «0f56� Ma81'/f..la"ClK�» as 4 xstp6ypaq>ou<; 'tOJlOU<;, 1tspi1tou 7000 crsl..ios<;. EKo6crst<; 'tou PtPA.iou au'toi> tyt­ vav apyo'tspa cr'ttl Btve'tia JlS Tl1 po�9sta 'tOU Jl<l9TJJ.t<lnKoi> M1ta/..6.vou Bam/..61toul..ou .

:I:'ta ft6.wtva oioa�av Kat ot KspKUpaiot E1>ytvt.o<; Boi>J..yap1]<; (1716-1979) Kat Nt.K1]­ <popo<; @EOTOK'I]<; (1731-1800) 1tOU <J1tOUoacrav <J'ttlV Eupffi1tTJ <l>ucrtKOJ.ta9TJJ.tanK6. Kat <l>t'Ao­ croq>ia. Amoi �'t(lV oacrKa'Aot <J'ttl :I:xot..� Ayiou Opou<;, <J'ttlV Krovcr'tavnvo'61tOATJ Kat <J'ttl Pou­ Jlavia. ilioacrKav a1t6 �tva PtPA.ia 1tou sixav JlS'taq>pacrst at..Aa Kat a1t6 otK6. 'tOU<;. Ts'AtK6.

BJlEtVav cr'ttl Procria ro<; KATJptKoi. 0 lro<J1]<p Mot.<Jt.ooa;, J.t<l9TJ�<; 'tOU M1ta1..6.vou, cr1tol>oacrs cr'ttl BttvVTJ <l>tl..oaocpia Kat Ma9TJJl<l'ttKU. 'H'taV oteu9uv�<; <J'ttlV 16.crsto :I:xot..� . 'Eypa'lfE TllV <<Ano}.oyia» (1780) Kat J!S't6.q>pacrs �tva PtPA.ia.

0 A9av. 'l'allioa<; (1764-1829) ilteu9u­ v�<; 'tOU :I:xot..eiou cr'ta ftaweva, auyypacpta<; Kat JlE'taq>pacr�<; JlU9TJJ.lanKrov PtPA.irov.

:I:sl..ioa a1t6 'to PtPA.io «0f56� Ma811JlarzK�» Ta 1tEPteXOJ.ttva 'tOU PtPA.iou au'to6 sivat Keq>a.Aata (l1t0 't(l :I:'totXEia 'tOU EuK'AeioTJ, rsro­ JlS'tpia E1ti1teoTJ Kat :I:'tepEOJ.lE'tpia, TptyrovoJ.tS­ 'tpia, rsroypaq>ia JlS Tl1 Y*VTJ a<paipa K.a.

0 MnaJ.. avo<; Ba<Jt.l..ono'UI..o<; (1699-1765) KA.11ptK6<; Kat oacrKat..o<; ts11crs 40 XP6vta cr'ta ft6.wtva 01tOU oioacrKE Ma9TJJ.t<l'ttKU. 'Eypa'lfe Apt9J!TJ'ttKll Kat fEOOJ.lS'tpia Kat tyws oteu9u­ V�<; <J'ttlV l:xoJ..1] rKwi>pa yvrocr� EID<JTJ<; JlS n<; ovoJ.tacrie<; Mnal..avaia l:xoJ..1], MqaJ..11 EYKAEIAm A' 89 T.l/4

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H e�tll�'l Tov aGTepwv A(yyro Tll � acr'ta9tta� 'tO'l> atpiou, nou tivat 'to «crroJ.La» 'tou npro'tacr'tspa, 11 ouvaJ.LtldJ 2 Ka'tap­ ptua.., Tll � UA11� A(yyro papUTil't<l�, sxtt� ())� U1tO'tSAtcrJ.La TllV UU�11<ill Tll� 1t'l>KVO't11't<l� (j'tO KEv'tpO 'tO'l> npro'tacr'tspa. Mt aUa A6yta, <j>UV't(l(J'tti'tt eva 'ttpacrno xrovi 'tO onoio K<l'tUATJ"(tt crt eva Koupa. To xrovi tivat ytJ.Lci'to an6 ptucr'tij nAacr'ttAiVll 11 onoia A(yyro papi>Til'ta� pstt npo� 'tov 1tAacr'ttAiV11 J!Scra cr'to Koupa 'tocro ouva'ta rocr'tt 11 1tAacr'tt­ "-VlJ•IJu.. 'Eva AtV11 an6 uypij yivt'tat cr'ttptiJ Kat 11 nita.., tivat 'tocro t­ crxupiJ rocr'tt OA11 11 nA<lcr'ttAiVll 11 onoia l>nTJPXt apxtKa cr'tO xrovi va X,ropcitt 'ttAtKa crt eva 1tOAU mo J.ltKp6 xropo 6nro� am6v 'tou Koupa. Au'tij tivat 11 �haOtKacria 11 onoia cruv'ttAti'tat crt evav npro'tacr'tspa. Q� tK 'tOU'tO'l>, o 1tl>pij­ va� 'tO'l> npro'tacr'tspa yivt'tat aOtaq>avfJ� (crTilV apxfl TJ't<l­ Vt aspto) Kat Otv SXO'l>J.lt (l1t00Atta tvspytta�. H Ota'tfJp11<ill Tll � tvtpytia� 011J.Ltol>pyti UU�11<J11 Tll � 9tpJ.LOKpacria� Kat Tll � 1tlt<J11 � Kat J.Lt <l'l>'tO 'tOV 'tp01t0 (nap'tt J.ll<l pa9ta avacra . . . ! ), 11 uopocr'tanldJ tcropponia tntPpaouvtt Tll � Ka'tappwa.., Tll � J.La�a� npo� 'to tcrro'ttptK6 Kat apxi�tt 11 ppaotia crucr'toAij, <>11A<loij m «q>ptvaptcrJ.La» Tll � Ka'tap­ ptua..,� Tll � UA11� npo� 'tO Kmpo 'tO'l> acr'ttptou.

A� 'tO oouJ.Lt mo anA,a. q,av'tacr'tti'tt J.Lia 'ttpacr'tta 'toup'ta ytvt9A,irov 11 onoia sxtt noUa cr'tproJ.La'ta an6 croKoAa'ta, KpsJ.La, J.L1ttcrK6'ta, cravnyi> K'tA. To Ka'tro cr'tpro J.l<l (paa..,) tivat 'tO J.l1ttcrKO'tO Kat 'tO navro tivat 11 (j(lV'ttyU (KOpl>q>TJ ). 'E'tat 01t(J)� J.Lt­ 't<l<j>Sptt 11 J.l11'tSpa (j� Tll 'tOUp'ta yta va Tll paA€t (j'tO 'tp<l1tS�l Tll � q>Wytl an6 't(l xspta Kat 1tS<j>'ttl navro (j'tO 'tp<l1tS�t Ka9t'ta. Mt 'tO XW1tll J.l<l Tll � 'tOUp'ta� navro (j'tO 'tpans�t �tKtVatt J.ll<l crttpa an6 aA,umoro'ta ytyovo'ta. To cr'tproJ.La Tll � cravnyu cruJ.Lnte�t'tat navro cr'to cr'tproJ.La Tll KpsJ.La� Kat yi­ vov'tat eva, 01tO'l> J.lt T11 crttpa 'tO'l> cruJ.LmS�t'tat navro crTil crOKOAa'ta Kat "(tVOV'tat Ev<l Kat 'ttAtKa OA(l J.l<l�i (jUJ.L1ttS�OV'tat navro cr'tO J.l1ttcrKO'tO Kat K<l'tUATJ"(O'l>J.lt (jTil yvrocr'tij crKllvTJ 01tO'l> 11 J.l11'tSpa cra� cprova�tt yta 'tou� AtKsOt� cr'to craA,6vt Tll � Kat sxouJ.Lt J.Lia 'tOUp'ta 11 onoia nMov J.LOta�tt J.Lt nivaKa aq>11P11J.LSV11� 'tSXVTI� 6nou otv J.L1topti� va �txropicrtt� 'tO crxflJ.La'ta Kat 'ta XPOOJ.la'ta!

Tiro� OO'l>AtUtt AOt1tOV 11 uopocr'tanldJ nita.., ; q,UV't(l(J'tti'tt 'tO cr'tpOOJ.l<l Tll � crOKOAa't<l� va tivat nayro't6 croKoAa'ta (o11A,aoij mo <iKA11p6) 'tO onoio J.L1topti va crtaJ.La'tijcrtt Tll Piat11 Ka'tapptua.., 'tou cr'tproJ.La'to� Tll � cravnyoKpsJ.La� xropi� va yivouv 6/.,a J.La�i eva npayJ.La. Am6 aKptPro� Kavtt 11 uopocr'tanldJ nita..,, cruyKpa'tti 'tO'l>� navro «OpO<j>O'l>�». Eivat Kat 0 A(yyo� 1tO'l> Ot 'ttK'tOVtKS� 1tAa­ Kt� Tll � rij� Otv pouAta�ouv cr'tO cr'tpOOJ.l<l 'tO'l> J.La'YJ.l<l'tO� (A,apa) 1tO'l> ppicrKt'tat an6 Ka'tro 'tO'l>�.

NOJ.l�ffi O'tt 'tropa 't(l 1tpayJ.La'ta nspa (l1t0 1tt0 ytucrnKa tivat Kat 1tto �tKa9apa, 'tO'l>Aaxtcr'tOV 01t'tt­ Ka. T0 ytyov6� O'tt Otv J.l1t0pti'tt va 1ttScrt'tt eva J.l1tO'l>KaAt J.lt vtp6 J.Lt 'tU xspta cra� Kat Va 'tO crnacrt'tt, tivat <l<j>SVO� AOyffi 'tO'l> 1tA<l­ cr'ttKOU 'tO'l> J.l1tO'l>K<lAtoU Kat aq>t'tspou A6yro Tll � uopocr'tanldJ 1tlt­ <ill� mu vtpou cr'to tcrro'ttptK6 mu.

A.pa, Ka9ro� crucr'tslli'tat o acr'tspa� au�avt'tat napill11Aa 11 9tpJ.LOKpacria 'tO'l>. :ETil cruvsxtta A(yyro Tll � UU�11<ill� Tll � 9tpJ.LOKpa­ cria� o acr'tspa� anoK'tcitt aKnvop6A0 1tl>pijva, 011Aaoij 11 tvspytta 'tO'l> J.Lt'taOiOt'tat J.Lt J.LOpq>ij aK'twopoA,ia�. Au'tij 11 OtaOtKacria cruvt­ X�t'tat sro� O'tO'l> 11 9tpJ.LOKpacria 'tO'l> 1tl>pijva vu �t1ttpacrtt 'tO'l> 106 pa9J.LOU� Kelvin iJ 'tO'l>� 999726,84 pa9J.Lo'6� Ktcriou, on6'tt Kat apxi�ouv 0t npro'tt� 9tpJ.l01tl>p11VtKS� UV'ttOpacrtt� (Kaua..,) 'tO'l> uopoy6vou.

2 dUVUJ.tUcrt f:VVOel't l J.tlU KU'tclCTtUcrrt fl paOSlYJ.1U sivat 11 'tUxQ'tTI'tU.

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Ms Tl1 1tapa1t<ivro �ta­ �tKacria, TJ 01toia �tapKsi EKa'tOJlfl'Upta XP6vta, Ka­ 'taA.i)youJlE O"'tOV aya1tT]'t6 Jla<; 'HA.to, o o1toio<; esro­ pshat tva acr'tept JltKpou J.l.Eyeeou<;! Av Kat 'tO «Jlt­ Kp6», 6crov acpopa 'ta JlE­ VY Canis yeeTJ 'tOU crUJl1taV'tO<;, EtVat Majoris Ka'tt 'tO Q'XE'ttKO 01tro<; O'U­ vi)et�E va Mst o edo<; AA.­ pep'to<;. 0 'HA.to<; a1tO'tEAEi 'tO 99.8632% Tll <; O'UVOAt­ Ki)<; Jln�a<; 'tOU T]AtaKOU O'UO''ti) Jla'tO<;. fta va 1tapOUJlE Jlia tOEa 'tOU 1t00'0 «JltKpO» EiVat 'tO acr'tpO 'tOU T]AtaKOU Jla<; O'UO''ti)Jla'to<;, a<; 'ta�t�E'JIOUJ.l.E JlEptKa par­ sec(1 parsec= 3,26 ETll q>ro't6<;3) JlEO"a cr'to axav€<; �t<iO"Tll Jla ro<; 'tOV acr'tsptcrJ.1.6 'tou Meya Kurova.

D1tro<; 1tapaTll pEhE 1tpOKEt'tat yta tva «JlEY<iAO>> aO''tEpt 01tOU 0 'HA.to<; Jla<;, �i1tA.a 'tOU ea E­ J.l.Ota�E 01tro<; tva JlUPJlTJYKt Jl1tpOO''ta O'E Jlta cpaA.atva. A<; JlTJV avaq>EpOUJlE 1tro<; ea cpav'ta�E TJ fT] Jl1tpOO''ta 'tOU. To 6voJla amou 'tou acr'tsptou Eivat VY Meya<; Kurova<; (Canis Majoris) Kat 'tov cruvo�€Uouv J.l.EptKa svro1trocrtaKa cr'tancrnKa cr'tmxEia: [!] BpiO"KE'tat 'tov acr'tsptcrJ.1.6 'tou MsyaA.ou LKl>A.ou, 5000 ETll cpro't6<; JlaKpta. [!] 'Exst 2,100 <pop€<; 'to J.l.Eyseo<; 'tou 'HA.tou Jla<;. [!] Av 'tOV pa�aJlE O''tO KEv'tpO 'tOU T]AtaKO'U Jla<; O'UO''ti)Jla'tO<; ea �E1tEpVOUO'E TllV 'tpoxta 'tOU Kp6vou. [!] To <pro<; eeA.Et 1tEptO'O'O'tEpo a1t6 8 ropE<; yta va �tacrxicrst TllV 1tEptq>EpEta 'tOU. AvaA.oytO''tEhE O'tt 'tO <pro<; a1t6 'tOV 'HA.to yta va q>'taO'Et O'Tll fT] XPEta�E'tat 1tEpt1tOU 8 AE1t'ta Tll <; ropa<;. [!] Au't6 sivat Kat 'tO J!EyaA.u'tE­ po acr'tpO 1tOU yvropi�OUJ.l.E aUa o faA.a�ia<; Jla<; 1ttea­ v6'ta'ta EXEt Kat aA.A.a acr'tpa aKOJlTJ JlEyaA.utspa ta o1toia 6Jlro<; Kpupov'tat 1ticrro a1t6 aepta Kat O'KOVTJ Kat �EV Jl1tOpOUJlE va 'ta �OUJ.l.E, OXt aKOJla 'tOUA<iXtO''tOV. [!] AA.A.a 6cro 1tto JlEyaA.o<; o a­ crtepa<;, t6cro mo «'JIUXPO<;».

A<; 'tOV eauJlaO'OUJlE Kat a1t6 Jlia <iMTJ 01t'ttKTJ yrovia (TJ JltKpi) KouKi�a cr'ta aptcrtspa sivat o 'HA.to<; Jla<;):

1 tTo<; <pon6<; sivat 11 a1t6crTa<nJ 1tou Otawst To <p� J.t£LU an6 T�iot sv6<; sTou<; J,Jicra crTo otaO'tll Jla. Auri] ma<nJ sivat 1tSpt1tOU SVVlUJllcrt TptOSKUTOJlJlUplU ;&.t6J.1stpa.

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Ila.pa.Ka:tro oivovtat JJ£ptKOi E�tl1tVot 'tp61tot ypinoprov Kat «iJJ£crrov t>1toA,oytcrJ.1COV, yta Vll EV'tU1trocrtcil;E'tE J.1S n<; . . . tl1tOMYyt<J'ttKE<; cra.<; tKet.V6tTJ'tE<;.

llOAA0.1tAO.O"to.O"f.lO� Kci9t a.pt9f.lOU f.lE 'tO 1 1 fta. 1ta.pciOEtYJ..lll 51236 X 11 -k'tllV llPXJl ypciq>OUJJ£ 'tOV a.pteJ..to J..l€ Evil J..lll OEVtKO <J'tllV a.pxfl cra.v 1tpono 'l'llQ>lO 051236 -Xropil;OUJJ£ VOlltcl tll 'l'llQ>tll tO'U 1tllpll1tclV(l) a.pteJ..tou 0 5 1 2 3 6 -AcpiJvouJJ£ to tsA.euta.io 'l'llq>io (to 6°) ro<; txst Kilt KtvOUJ..lllcrtE ll1t0 tll Oe�tci 1tpo<; tll a.ptcrtspci, 1tpocrettovta.<; a.vci ouo ota.ooxtK«i ta. 'lfllq>ia., Kaes 'l'llq>io 011A.a.oiJ J.1S to yettovtK6 tou.

Ta. a.1totsA.tcrJ..ta.ta. to1toestoi>vta.t OtllOOXtKcl. L\11Ail0TJ : 0 5 1 2 3 6 (0+5) (5+1) (1+2) (2+3) (3+6) 6 5 6 3 5 9 6 -TEAtKcl 51236 X 11=563596.

(Av U1tapxst Kpa.tOUJ..lEVO , tots J.lSta.q>tpouJJ£ to KplltOUJ..lEVO <J'tll <; E1tOJ.lSV11<; tcl�ll<; 'l'llQ>lO ) .

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(4_5) IIpocres'tOt>JJ£ 'tou<; ouo a.pte,..ou<; 4+5=9 To1toee'tOUJ.lE 'tO et.1tO'tEAecrJ.la. a.vciJJ£cra. cr'to 4 Kat cr'to 5 (4_9_5) TsA.tKci: 45 X 11= 495 • Av 'tO aepotcrJ..ta. 'trov ouo 'lfllq>irov siva.t J.lSYaAU'tEpo 'tOU 10, 'tO'tE 1tpocres'tO'UJ.l£ 'tO Kpll'tOUJ..lEVO cr'tO 1tpono 'l'llQ>tO yta. 1tet.pciOEtYJ..lll 0 a.pteJ.16<; 67.

(6_7) 6+7=13 fpaq>OUJJ£ 'tO 3 Kat (1+6=7) (7_3_7) IlpciyJ..ta.n:

Tiro� 9a U'JI<l>O"tt� a'to n:'tpciyrovo K«i9t <>nvftqno apt9f.l6 1tOt> 'tti..Etrovtt o-t 5 fta. 1ta.pciostyJ..ta. to 35.

- Ilcips to 'lfllq>io 'tll <; OEK«ioa.<; Kat a.u�11crt to Klltcl 1. ( 3+1=4 ) - IIoUa1tA.acria.crs tov a.pteJ.16 1tou PPTJKE<; (4) J.1S to 'lfllq>io 'tll <; OEK«ioa.<; 3x4=12 - r Pcl'lf€ tO ll1t0tSAecrJ..lll Kat Ot1tAa tO 25 (1225) 2 35 =1225 firo� 9a tl'JiroO"tl� (;'tO TETpayrovo K<ifh; ()l'JIJt(jHO apt9f.lO fta. 1tllpcl0Et'YJ..lll to 36.

- I:KE'IfOU 'tllV 1tA11crltcrtEp11 OEK«ioa. crto 36.

- Eiva.t to 40 sivat 4 J..lOVcios<; 1tcivro a.1t6 to 36.

- Tropa. ea. KlltEPsi<; ll1t0 tOV llPXlKO a.pteJ.16 6cre<; J.lOVclOE<; a.v€P11KE<;, 'tO 36 ea. yivst 32

EYKAEIAm A' 89 T.l/9

------ AptOpl)TlKQ TplK KQl \l1tOAoytGpOi GTO........ ypTjyopa!!!!!! ------

- Nospci 1tOAAa1tA.acrtci�st<; 32x40, 'tO Kcivst<; ro<; s;f]<; : • Ilpona 1tOAAa1tAaO"Ul�€t<; 'tO 32 f.l€ 'tO 10 32x10 =320.

• 'E1t€t'ta 1tOMa1tAacrtci�€t<; 'tO a1tO'tE-AsO"f.la f.l€ 'tO 4. 0uf.lf]crou on yta va 1tOMa1tA.acrtcicrst<; f.l€ 'tO 4, bl1tAacrtci�st<; 8uo <pops<;.

• �l1tAacrtci�st<; f.lta <popci 320+320=640.

• To bl1tAacrtci�st<; ;avci 1280.

- Kpa'tclf.l€ 'tO 1280.

-To 36 f:ivat 4 f.lOVcibs<; «f.lUKptci» a1t6 'tO 40.

-Y\j/(OV€t<; cr'to 'tUpciyrovo 'tllV 8ta<popci 4, 4x4=16 -Ilpocres't€t<; crm 16 'tO 1280 1280+16=1296 36"2=1296 -TsA.tKci Ms s;acrKllO"ll yivs'tat vospci, xropi<; f.lOAu�i Kat xap'ti. rta va 1tOAAa1tAaO'tclO'€l>; Jl€ TO 5 ApK€i va 8tatp€crst<; 'tOV aptef.l6 f.l€ 'tO 2 Kat f.l€'tcl va 1tOMa1tAacrtcicrst<; f.l€ 'tO 10.

5x13 13/2=6.5 6.5x10=65 rta va 1tOAAa1tA«O'tclO'Et� JlE TO 9 ApKsi va 1tOMa1tA.amcicrst<; f.l€ 'tO 10 Kat Ka't01tlV a1t6 'tO a1tO'tEAsO"f.la va a<patpEcr€t<; 'tO i8to 'tOV aptef.l6.

9x45 10x45=450 450-45=405 Y 1tOAO'YtO'J10�

K'l)fJtKfl� pi�a� Ilro<; ea aa<; <patVO'taV va U1tOAoyi�a't€ 'tllV KU�tKi] pi�a sv6<; aptef.lou crs A.iya b€U't€p0A€1t'ta. AV f.lll 'tt clMO ea i]'taV s;atpsnKci svro1trocrtaK6. Ilap on f.lotci�st s;atpsnKci 8ucrKoAo, gsv f:ivat. Av 'tO KclV€'t€ f.l1tpOO"'tcl 0"€ <piAou<;, criyoupa ea KAE'If€'t€ 'tllV 1tapcicr'taQ"11 .

Zll'tf:icr't€ a1t6 €vav <piAo aa<; va €1ttA.E;st 01totOVbi]1tO't€ aptef.l6 a1t6 'tO 1 flSXPt 'tO 100 xropi<; va aa<; 'tOV a1tOKaAU'If€t , O"'tllV ()l)VEX€ta va XP110lf.l01toti]crst T'llV aptef.lOf.lllXaVil 'tOU KtVll'tOU 'tOU Kat va 'tOV U'lf000"€t O"TllV 'tPlTll 8uvaf.1.11· Na aa<; avaKotvrocrst 'tO a1to't€AscrJ.la ·

Kat scrsi<; aJ.lscrro<; va �pshs 'tOV aptef.l6. T11v KU�tKf] pi�a 'tOU a1tO't€AEO"f.la'tO<;. �€h€ 1tro<; ea 'tO KclV€'t€, apXtKcl ea 1tpE1t€t va f.l1tOpsi'ts va a1tOf.lVllf.lOVrncrs'ts 6A.ou<; 'tOU<; K6�ou<; a1t6 'tO 1 flSXPt 'tO 10. ·

13 23 33 43 53 63 73 83 93 103

1 8 27 64 125 216 343 512 729 1000

IlapaTll pOOV'ta<; 'tOV 1tapa1tcivro 1ttVciKa 8tamcr'trovouf.1.€ on 6A.a 'ta 't€AtKci 'lfll<pia 'trov K6�rov sivat 8ta<popsnKci apa criyoupa yvropi�OUf.l€ 'tO 'tsAf:maio 'lfll<pio T'll<; KU�tKi]<; pi�a<;. Ilapa'tll pouf.l€ s1tiQ"11<; on s;atprov'ta<; 'ta 2, 3, 7 Kat 8 'tO 't€AsU'tatO 'lfll<plO 'tOU JC6�0U Kat 'tOU aptef.lOU 1tOU 'lfclXVOUf.l€ sivat 'tO i8to.

A<; bOUf.l€ (j'\)YK€Kptf.1.Eva 1tapa8siyfla'ta, 1tapaKaA.f:hs €va <piAo aa<; va crK€<p't€i €vav aptef.l6 a1t6 'tO 1 flSXPt 'tO 100, 'tO KclV€t, O"T'IlV ()l)VEX€ta 'tOV U'lfOOV€t f.l€ 'tO KOf.l1tlOU't€pclKt O"TllV 'tPlTll bUVaf.lll Kat aa<; avaKOtVOOV€t 'tO a1tO'tEAsO"fla 1t.X o aptef.l6<; 250047, 'tO 't€Aro'taio 'lfll<pio 'tou aptef.loU sivat 'tO 7, Kot'tclf.l€ 'tOV 1tapa1tclV(J) 1ttVaKa Kat bta1tlO"'tOOVOUf.l€ 'tO't€ TO 'tEAE'l)Taio \Jill (j)lO Tll � K'l)fJtKfl� TO'l) pi�a� dvat TO 3. l:T11V ()l)VEX€ta ayvoOUf.l€ a1t6 'tOV aptef.l6 'ta 'tpia 't€AsU'taia 'l'll<ria, o 250047 yivs'tat 250, Kot'tclf.l€ ;avci 'tOV 1tapa1tclV(J) 1ttVclKa 'tO 250 �ptcrK€'tat aVclf.l€O"a O"'tO 216 Kat 'tO 343 apa 'tO 'lfll<ptO 1tOU J.la<; €Vbta<pEp€t �picrK€'tat aVclJ.l€O"a O"'tO 6 Kat 'tO 7. Ilciv'ta €1tlAE'Y01>f.l€ 'tO f.ltKpO't€pO aptef.lO €V 1tpOK€tflEv(J) 'tO 6. Apa 0 �ll TOUJlEVO� apteJlO� dvat TO 63.

'Eva 7tapci8styf.la aKOf.lll· Na �pse€i 11 KU�tKi] pi�a 'tOU 704969. A1t6 'tO 't€Aro'taio 'lfll<pio Ka'taA.a�aivoufl€ on 'tO 't€A€maio 'l'll<rio T'll<; KU�tKi]<; pi�a<; sivat 'tO 9. AyvooUf.l€ 'ta 'tpia 't€Aro'ta{a 'lfll<pta 'tOU 704969 apa flEv€t 704. A1t6 'tOV 1tapa1tclV(J) 1ttVaKa 'tO 704 �ptO"K€'tat aVclf.l€pa O"'tO 8 Kat O"'tO 9, 1tatpVOUf.l€ 'tO f.ltKpO't€pO, apa K'l)fJtKf] pi�a dvat TO 89. (!:'l)Vf:X�E't'(ll ) •••

EYKAEIA� A' 89 T.l/10

MOVT[A01TOti]OI]

KO I

[�tOWOI]

=======

Ito npwto 6taywvtotJa n�pa 40

11011cl:6€( ICQI crtO 6£Ut£po 30 f.IOVcl:6£c;.

IuvoAtiCQ �pa 40+30=70 f.IOVcl:6Ec;.

I to npwto 6tayWIItolla m\pa 35

j.l0VG6£<; IC(U crtO MUUpo 45 j.l0VG6E<;.

IUY,OMKQ n�pa 35+45;:80 jlOVa6Ec;.

xw €50. Av ayopaow eva �euyapL Extu €35. Av ayopaow E:va nanOll'tOLa nou otOLXL�OUV €35, ea I'")A::xrpovLKO illaLXVLOL nou £xw€(50-35}=€15.

OLOLXL�EL€20, ea EXLu €(_,5-20)=€15.

Apy'6p1J Ila.va.yl<f>Ta.

Ito pt�to lita�·t�VIOilO. r.�pa o. ovcill£c /({H OTO 6£uupo � f!Ovli6£�.

Iuvol,t�a rt�pa o. +3 �tovaOEt;.

Exw€x. Av ayopaow £va Ka1tEAO 1tOU otOLX�El f¥, ea £xw€(x-y).

•

rta va. a.va.xa.pa.atiJaouJ-1£ J..ltO. xoaonrta 1tO'l> J.lB'tO.�CtMt'tat, 6xroc; tiva.t TJ TJAtKia., 'tO uwoc;, 11 'tO.xU'tTJ'tO., m �a.9J.loi a-ra J.la9iJJ.la'ta K'tA., XPTJatJ.l01tot0'6J.lt ypaJ.lJ.lO.'ta iJ aUJ.l�oA.a, -ra. oxoia.

Myov-ra.t fJET«Pl1JTE<;.

•

E;iaroG1J fl£ tva.v O.yvromo ovoJ.lcist-ra.t J.lta tcrO'tTJ'tO. ,xou xtptEXtt a.pt9J.lo'6c; Kat €va. ypUJ.lJ.lO. ( ciyvroa-roc;)

•

A'6G1J T1]<; t;iaro(J1]<; tiva.t o a.pt9J.l�, xou 6-ra.v a.vnKa.-ra.atiJcrtt -rov ciyvroa-ro, t1tO.ATJ9t'6tt 'tTJV tcrO'tTJ'tO..

EYKAEIAHI: A' 89 -r.l/11

------- MOV'rt:J..onoiqaq KUl .:;t<Jroaq

-------

IKtljlou tvav ap181-16 01T6 TO 1 ·25

AcpalpEOE TOV ap181J6 1TOU OKt<pTilKEc;

Na O'Kt<p'tdtt tvav a/J...o apt6f..l6 Kat va a.KoA.ou6i)crt't£ 'tO. 1tto xavro PiJJ..la'ta.

Na t�rryi)crt't£ ytati 11 axaV'tll O'll dvat xav-ra 7.

3.

E�wtim1:1.; !UH 1TpO�AlhHJTa A ) Lt tva tcrocrKtAt� -rpiyrovo 11 PaO'll tivat OtxMima Kat Ka-ra 3 J..t£yal.:6-rtp11 axo n� icrt� xA.cup€� 'tOU. a) Na ypO.'Ift'tt 'tllV a.A.ytPptKi) xapacr-ra.O'll xou tK<ppa�tt 'tllV xtpiJ..lt-rpo -rou.

'Ecr-rro x Jlia axo n� icrt� xA.cup€� -rou, -r6n: 11 PaO'll -rou 6a tivat : 2x +3 Apa 11 1ttpi!l£-rp6� -rou : x+x+2x+3=4x+3 t3) Av 11 1ttpi!l£-rpo� tivat 15cm, xow� tivat ot Otacr-racrtt� -rou tcrocrKtA.ou� -rptyrovou 4x+3=15 =>4x=15-3 =>4x=12 =>x=12:3=4 cm Ka6t Ilia axo n� icrt� xA.cupe� Kat 11 PaO'll -rou 11cm. m

To KOO''tO� yta X 0£U'ttp6At1t't0. O'UVOJ..ltAia� J..lt tva 'tll M<provo tivat 0,5 tupro O.U�11J..lf.v0 KO.'ta 10 tupro xayto KOO''tO� O'UVOJ..ltAia�. II6cra 0£U'ttp6At1t't0. OJ..ltAia� O.V'ttO''totXOUV 0'£ J..l11VO.t0 A.oypaptcrJ..l6 40 tupro; 0,5·x+ 10 = 40=> 0,5·x= 30=> x = 60 sec r) To 1t11AiKo -rou -rt-rpaxA.acriou tv6� apt6J..lOU J..lt -ro 5 tivat 16. Iloto� tivat o aptJ.16� am6� ; 4·x = 16=> 3·X= 80=>X= 20 5 -

B•IJI..•oypa(j)ia : LXOAtKO PtPA.io A ruJ..lVO.O'tOU ,EVO'tll 't0. 1 P.(t0'0'tll 't0. -t�tO'(I)O'll ) .Yxoupytio Ilatotia� Kat IIoA.tncrJ..lou .IIatoayroytK6 Ivcrn-rou-ro Kuxpou. Y1t11Ptcrtia Avaxro�11� IIpoypa.J..tJ..la-rrov.

EYKAEIAm A' 89 -r.l/12

M�:OolioAoy la EnL\uoq �

npopAq iJaTu;v - - -c

Enetofl oev '1>1tUPXet fJlU «(J'\)Vtarfl>> iJ eva� evtaio� aA.y6ptefJo� nou va Mvet 6/..a ta npo�/..fl fJata, eivat XPllcrtfJO, avti va KavoufJ.f: roxa� m<E'Jie�, va XP11<JtfJ01tOtoUfJe fJia J1t9oool..oyia aKo/..ouerovta� KU1totU P11JlaTa ta onoia ea fJU� OOTJrflcrouv 1tl0 eUKOAU <JtT] M<JT] tOU 1tpO�AllfJUtO�. 'Etcn xropi­ �OUfJe tT] otaOtKacria eni/..U <JT]� tou npo�/..fl fJato� cre 4 �llfJUtU, 01tffi� 1tUpUKUtffi.

I ) Avayvropurq otooptvrov, �llTO'Upivrov TO'U 1tpopl..l) J1aTo�, Ka9ro� Kat 'Pf.lclO'EIDV «WlOlcl», no'U Kpvpo'Uv Jla911JlaTtKt� tvvott�.

2) Avai..'V 0'1J Trov ot3oJ1tvrov, �1JTO'VJ1tvrov Kal q»paatrov «mlola».

3) :r.vv9t0'1J 4) Enal..� 9t'V0'1}-aVaGKOX1}0'1}

Eq>apfJoyE� tT]� 1tUpU1tUV(I) OtUOtKacria� ea KUVOUfJ.f: fJf: OUO XUPUKtT]ptcrttKa 1tpO�AllfJUtU. 1 1 poj}AIWU 1 "

0 Yeffi1tOVO� fJtU� 1tept<pEpetU� e1tl<JKE1ttetat eva XffiptO Kaee 1 0 fJEpe�, 0 KtT]VtUtpO� Kaee 1 2 fJEpe� Kat o oacraPXTJ� Kaee 1 5 f..l.Ep e�. Mia 11fJEpa (J'\)Va­ vtfJST]Kav Kat ot 3 fJ�i crto xropt6. a) MetU U1t0 1tO<Je� fJEpe� ea �UVU(J'\)VUVtT]eouv yta npffitTJ cpopa; p) II6cre� q>op£� ea EXet 1tUet 0 Kaeeva� tOU� crto XffiPtO, OtUV �UVU(J'\)VUVtT]eOUV yta 1tpiDtT] q>Opa;

A \lcnl Bl)�w 1ff)O>TO . AvayvroptO'I] otooptvrov, �llTO'UJli­

vrov Ka9ro� Kat 'Pf.lclO"trov «ilitota» no'U Kp'6po'Uv Jla911JlaTtKi� MOl£�. �ta�a�OUfJe npocreXttKa, noW� cpop£�, to np6�A11fJU yta va KatavoflcroufJe nota eivat ta oeoof..l.E­ va, nota ta �11tOUfJ.f:VU Kat note� ot q>pacret� «ilit­ ota» ta onoia unoypUfJfJi�oufJ.f: navro crto npo�/..11 f..l.U il tU KUtaypaq>OU fJ.f: crto xapti fJU�. LtO 1tp6�A11fJU fJU� eivat: AEOO�{:vu : 0 Yeffi1t0VO� e1tlcrKE1ttetUt to xropt6 KU­ ee 1 0 fJEpe�.

0 KtT]viatpo� e1ttm<:E1ttetat to xropt6 Kaee 1 2 fJEpe�.

0 oacraPXTJ� emm<:£ntetat to xropt6 Kaee 1 5 f..l.Epe�.

«A£�11 - ilitoi» eivat 11 M�11 'Ka9t'» ZllTOVJleva:To tpcl>TI]Jla a) fJf: q>pa<JT] «ilitOi» yta 1tf.lcOTI] 'POf.lcl.

TO Ef.lcOTI]Jla �)

TI.

K1>pavac; - K. I:.al..ap1Jc;

Bl'J !J U i h:(m:p o (TO O'I]JlaVTlKOTtpo P11Jla): ava­ A'UO'I] OEOOJlivrov, �11TO'Upivrov Kat 'Pf.lclO'trov «ilil.Ota» Ava/..u<JT] <JT]fJUtVet va KatavoiJcroufJ.f: note� fJUST]­ fJUttKE� evvote� eKq>pa�ouv ta oeoof..l&va, ta �11tou­ fJeva Kaec.o� Kat ot cppacret� «ilitota».

Ava/..1> 0'1] otOOJlivrov. Lto np6�/..11 fJU fJU�, o yero­ n6vo� emm<:£ntetat to xropt6 Kaee 1 0 fJEpe�, <JTJ­ f..l.Uivet 6tt to e1tlm<:E1ttetat aptefJo 11f..lepc.Ov noUa­ n/..acrto tou 1 0, o KtT]viatpo� noUan/..acno tou 1 2 Kat 0 oacrapm� noA/..an/..acno tOU 1 5 . LUVe1tiD� ea (J'\)VUVtiDVtat Kat Ot 3 fJU�t <JtO XffiPtO, <JtU KOlVcl 1tOlla1tJ..clO'ta tffiV aptefJIDV 1 0, 1 2, 1 5.

A vuhlml

�llTOlllJlS(!) \'

o.) EnetOfl to np6�A11fJU fJU� �11tclet va �pOUfJ.f: note ea �UVU(J'\)VUVtT]eOUV yta 1tpiDtT] q>opa, KUtaAa�Ut­ VOUfJe Ott fJU� �11tclet VU �pOUfJ.f: to 1tf.lcOTO KOlVO 1tOlla1tJ..clO'lO tffiV aptefJIDV 1 0, 1 2 Kat 1 5 1tOU etVat to E.K.Il tou�. f)) Enetofl EXOUfJe �pet uno to epiDtT]fJU a) fJ.f:tU U1t0 1tO<Je� fJEpe� ea �UVU(J'\)VUVtT]eouv yta 1tpiDtT] cpopa Kat ot e1ttm<:E'Jiet� yivovtat Kaee 1 0, 12 Kat 1 5 fJEpe� oev ExOUfJ.f: napa va OtatpE<JOUfJe to E.K.II fJf: tOU� aptefJOU� 1 0, 12 Kilt 1 5 .

B1hw Tp iTo : I:YN9EI:H

Luvee<JTJ <JT]f..l.Uivet va eKte/..EcroufJ.f: tt� npa�e� Kat va �pOUfJe tT] M<JT] tou npo�/..fl fJato� crUfJq>mva fJf: tT]V ava/..u <JT] nou £xoufJ.f: Kavet napanavro, 011/..aoiJ yta to np6�A11fJU fJU� va �pOUfJe to E.K.II Kat va to otatp£croufJ.f: fJf: tou� aptefJou� 1 0, 1 2 Kat 1 5 .

To P11Jla T'l� 0"6v9t0'1]� a'Pl}VtTat mov avayvro­ O'T'l · Ti:w.pTo

� i nw : ETIAAH9EYI:H ANAI:KOllHI:H Lto �iJfJa auto yivetat enaA.fleeu<JT] tou npo�/..fl fJa­ to� Kat e�eta�OUfJ.f: n. x i.

Av XPTJ<nfJonmfle11Kav 6/..a ta oeOOfJeva Av 11 M<JT] tou npo�/..fl fJato� eivat ')...oytK:fJ ii.

Kaero� Kat U')...').a .. epffitflfJUta ava/..oya fJf: tT] q>U<JT] tOU 1tpO�AllfJUtO�.

II.x crto np6�/..11 fJU fJ� 11 enaA.iJeeu<JT] fJ1tOpei va yivet fJf: to va �pOUfJe avaA.uttKa to E.K.II Kat va fJ.f:tpflcroufJ.f: tov aptefJo trov noA/..an/..acrirov to>V 1 0, 1 2 Kat 1 5 nou eivat nptv uno to E.K.II yta va �poufJ.f: n6cre� q>op£� EXet entm<:eq>tei o Kaeeva� to Xffipt6 fJEXPt tT]V 11fJEPU 1tOU (J'\)VUvtfJST]KaV K(lt Ot 3 yta npffitTJ cpopa.

EYKAEIAIU: A' 89 -r.l/13

-

------ M�:Ooool.oyia EniA.tlmJ� TipopA.qpO:Trov IIpopJ.:rtpa 2° 'Evac; aveonro/.:rJc; sxst 48 K61CK1va, 40 acmpa Kat 1 6 Khptva yapi<pal..a . a) II6crsc; TO noA.u opot6pop<psc; aveooscrpsc; pnopd va <pna�st ps au'ta TU yapi<pal..a ;

fl) An6 n6cra yapi<paA.a Ka9s XPcOJ..La:toc; ea EJ..shat KU9S UV900E<JJ..LfJ ;

U1to't

Bl]pa nproTo : avayvropurrt osoof.1tvrov, �fiTovpt­ vrov Km. <ppaasrov «mtoul.»

Asooptva: 0 aveonOOATJc; sxst 48 K6KKtVa, 40 cl0'1tpa Kat 1 6 KhptVU yapi<paAa. ZfiTOilptva: a) II6crsc; TO noM opot6pop<psc; aveooscrpsc; pnopsi va <pna�st; Ms <ppacrstc; «KAstota» i) TO no/..il ii) OpOtopop<pE� p) An6 n6cra yapi<paAa Kaes XPOOJ1U'toc; ea sxst Kaes aveooscrpTJ ; Bl] pa osilTspo : ANAAYI:H osooJ1Svmv, �TJTOups­ vmv Kat <ppacrsmv «KI..stoui>). .

Opotopopqu::� (JT)paivst, Ot aveooscrpsc; pE'tU�U Touc; va nspffixouv Tov ioto aptep6 K6KKtva, Tov i­ oto acrnpa Kat 'tOV ioto Khptva yapi<pal..a . To 1tOA.il (JT)paivst TOV psyaMTspo aptep6 opot6pop<pmv aveooscrprov nou pnopsi va <pna�st o aveonroA.TJc;. a) Enstoi) sxou11s va jlOtpacroujls Ta 48 K6KKtva, 'tU 40 acrnpa Kat 'tU 1 6 Khptva yapi<paAa O'S OpOtO­ pop<psc; aveooscrpsc;, KU'tUAa�aivoups 6n 0 apte­ p6c; 't(J)V aveooscrprov ea Otatpsi Touc; aptepouc; 48, 40 Kat 16 OTJAaOTJ ea sivat KotVO� OtatpETfl� TO'V�.

To np6�ATJ!lU �TJ'tclEt va �poups To jlEyaAu'tspo an6 'touc; KOtVOV� otatpETEc;, npsnst VU �poups 'tOV M.K.A Tmv apteprov 48, 40 Kat 1 6 . p ) E<p6crov 't U yapi<pal..a jlOtpcl�OV'tat crnc; aveo­ oscrpsc; ps 'tOV psytO''tO Kotv6 OtatpETTJ M.K.�, osv sxoups napa va otatpscroups Touc; aptellouc; 48, 40 Kat 1 6 ps 'tOV M.K.A. I:YN0E:EH EIIAAH<">EYEH H O"Uves(JT) Kat TJ snaA.i)esu(JT) a<pi)vs'tat crTov

avayvroO'TT) .

IlpoTEtvopEVa 1tpo� l..ilCJf1 1tpopJ..l]paTa. IIpoPA.flpa 1 ° : 'EXOUjlE 1 26 'tE'tpclota, 1 1 2 pol..il �ta Kat 4 2 y6psc;. a) II6cra TO noM opot6pop<pa osjla'ta pnopoups va <pna�ou11s; p) II6cra 'tE'tpclota, pOAU�ta Kat y6psc; ea nspffixst To Kaes os11a; IIpopJ.11pa 2°: H Kajlnava ptac; srulJmac; xwna Kaes 6 A.s­ nTa, lltac; aUTJc; Kaes 9 l..snTa Kat lltac; TpiTTJc; Kaes 1 5 l..sn'ta. Av xwni)crouv 6/..sc; 11a�i crnc; 8 n.p. a) Ilota ropa ea �avaxwni)crouv 6/..sc; pa�i yta nproTTJ <popa; p) II6crsc; <popsc; ea sxst XTUnTJO'Et TJ Kcl9E KUjlncl-

------

VU !lBXPt TT) crnypi) nou ea �aVUXTUnTJO'OUV 6/..sc; pa�i yta nproTTJ <popa; IIpoPA.flpa 3°: �uo poTocruKA.sncrTsc; avaxmpouv an6 To ioto (JT)psto A O''tOV ioto op6jlO, npoc; TTJV iota KU'tsU9-uv(JT). 0 nproToc; Tpsxst ps 60 kmlh Kat o osi>Tspoc; ps 80 km/h Kat �sKtva 1 h apy6Tspa an6 Tov npro­ To. Es n6crsc; ropsc; an6 TTJ crnypi) nou �EKiVTJcrE o nproToc; Kat crs nota an6cr'ta(JT) an6 TO (JT)psio A o OSU'tEpoc; ea <p'tclO'St 'tOV npro'to; Enstoi) osv unapxst jllU (((Jl)V'tUYTJ)) iJ £vac; s­ Vtaioc; aA.y6pte!loc; nou va A.uvst 6/..a Ta npo�A.i)pa­ Ta, sivat XPTJO'l!lO, avTi va Kavou11s wxaisc; crKs­ 'JIEtc;, va XPTJO"tjlonotoujls 11ia pEOoooJ.oyia aKo­ A.ouerovTac; Kanota Pl]paTa 'tU onoia ea !lac; OOTJ­ yi)crouv nto sUKOAa O'TTJ AU(JT) TOU npo�ATJjlUToc;.

'E'tcrt xmpi�oups TT) OtaOtKacria sniAu(JT)c; 'tOU npo­ �ATJjlUToc; crs 4 �i)pa'ta, 6nmc; napaKaTm. 1) AvayvroptCJfl osooptvrov, �fiTovptvrov Tol> 1tpopJ..l]paTo�, Ka9ro� Kat <ppaasrov «KAEt­ otcJ.)), 1tO'V Kpilpovv pa9f1paTtKE� MOtE�.

2) Avai..'V CJTI Trov osooptvrov , �fiTOvptvrov Kat <ppaasrov «mtotcJ.)).

3) I:ilvOECJfl 4) E1taA.1]9EVCJf1-avaaK61tfiCJfl E<papjloysc; TT)c; napanavm OtUOtKacriac; ea Kcl­ VOUjlS JlS OUO XUPUKTT)ptO''tlKcl npO�ATJJlU'tU. IIpopA.flpa 1 o 0 ysmn6voc; ptac; nspt<pspstac; smcrKsnTs'tat EvU XWpt6 Kaes 1 0 pEpsc;, 0 KTTJVlU'tpoc; Kaes 1 2 jlSpsc; Kat o oacraPXTJc; Kaes 1 5 jlSpsc;. Mia TJpEpa cruvav'ti)eTJKav Kat ot 3 11a�i crTo xmpt6. a) MsTa ano n6crsc; pspsc; ea �avacruvavTT)eouv yta nproTTJ <popa; p) P6crsc; <popsc; ea sxst nast 0 Kae£vac; TOuc; O''tO xmpt6, 6'tUV �avacruvavTT)eouv yta nproTTJ <popa; ·

Ailaq

Bl]pa nproTo. AvayvroptCJfl osooptvrov, �Tt­ Tooptvrov Ka9ro� Kat <ppaasrov «mtota>> 1tov Kpilj}ovv pa91]paTtKt� tvvow�.

�ta�a�oups npocrsxnKa, noWc; <popsc;, To np6�ATJJ1U yta VU KU'tUVOTJO'OUpS nota stVat 'tU OS­ OOpEvU, nota m �TJTOupsva Kat notsc; ot <ppacrstc; «KAstotcl)) TU onoia unoypajlp�OUjlE navm cr'to np6�AT)pa TJ 'tU KU'taypa<poups O''tO xap'ti pac;. L'tO np6�ATJjlU 11ac; sivat: Asooptva: 0 ysmn6voc; sntcrKEn'tE'tat 'tO xmpt6 Kcl­ es 1 0 jlEpsc;.

0 KTTJVlU'tpoc; E1tlcrKEn'tE'tat 'tO xmpt6 Kaes 1 2 pspsc;. \ 0 oacrapxTJc; E1tlcrKEn'tE'tat 'tO X pt6 Kaes 1 5 pspsc;. «AE�TJ - KA.stoh) sivat TJ M�TJ ' � s' » Z1]TOVpEVa: TO Epc0Tflf.1a a) pE <ppcl(JT) «KAstob) yta 1tpiDTfl <popa.

TO EpcOTflpa �)

EYKAEIAHI: A' 89 T.l/14

� �

------ Mt9o«iol..oyia EnilllGfl� Dpopl.:ruux-rrov

B'll JHl OtUTtpo ('TO O'fiJlaVTlKO'Ttpo �11Jla) : ava­ A'UO't) OtOOJltvrov, �fiTOt>Jltvrov Kat (j)pclO'trov «Ki..ttota» AvaA.t><Jll <Jll Jlaivtt va KatavoijcrO\lflt nott� Jlae11JlattK€� MOtE� EK<ppasow ta od)OJlEva, ta s11tOU­ JlEVa Kaero� Kat m <ppacrtt� «l<Attota>>. AvaA.t>O't) OtOOJltvrov. �to np6�A11Jlcl Jla�, o ytron6vo� tnt­ O"K€1ttttat tO xropt6 Kaet 1 0 JlEpt�. At>TO <Jll !laivtt on to tntO"K€ntttat apteJ.16 fiJltprov noUanMcrto tOU 1 0, 0 K'tTJViatpo� notJ..anf..acrto tOU 1 2 Kat 0 OaO"UPXll � noiJ..anf..acrtO tOU 1 5 .

Aval..t>O't) �fiTOt>Jltvrov �\lVtnffi� ea O"UVaVtiDVtat Kat ot 3 Jlasi O"tO XOOpt6, crta Kotva noUanMcrta trov apteJ.lroV 1 0, 1 2, 1 5 . a) Enttoi] to np6�A11JlU Jla� s11tatt v a �pOUJ.lt n6tt ea �aVaO"UVaV'tTJeouv yta npiD'tTJ <popa, KataJ..a�ai­ VOUJ.lf: 6tt Jla� sfltUtt va �pOUJ.lf: tO npcOTO KOLVO nollanl..aO"LO trov apteJ.lrov 1 0, 1 2 Kat 1 5 nou tivat tO E.K.II tOU�. �) Enttoi] txouJ.lt �ptt ano to tpiD'tTJJ.la a) Jltta ano n6crt� Jl€pt� ea �avaO"UVaVt11eouv yta npiD'tTJ <popa Kat ot tmcrKB'Iftt� yivovtat Kaet 1 0, 12 Kat 1 5 Jl€pt� otv txouJ.lt napa va otatp€croUJ.lt to E.K.II J1t tOU� apteJ.lOU� 1 0, 1 2 Kat 1 5 .

B11Jla tpito : l:YN9El:H �uvet<Jll <Jll J.laivtt va tKttA.ecrouJlt n� npa�tt� Kat va �pOUJ.lt 'tTJ M<Jll tou npo�A.iJJ.lato� crUJ.l<prova J1t 'tTJV avaA.uO"TJ nou txouflt Kavtt napanavro, o11A.aoi] yta to np6�A11JlU Jla� va �pOUJ.lt to E.K.II Kat va tO OtatpBO"OUJ.lf: J1t tOU� apteJ.lOU� 1 0 , 1 2 Kat 1 5 .

To �llJl« Tfl� O'Uv9t0'1)� «(J)llVtTat O'TOV uvuyvro­ O"Tff ·

TETUpTO �llJlU: EllAAH9EYl:H - ANAl:KO DHl:H

-

�to �iJJla auto yivttat tnaA.i]ew<Jll tou npo�A.iJJ.la­ to� Kat t�ttUsOUJ.lf: n.x Av XP11crtJ.lOnoti]etKav 6'Aa ta otooJ.l€va m. iv.

Av 11 M<Jll tou npo�A.i]Jlato� tivat A.oytKi] Kaero� Kat atJ..a tpOO'tijJ.lata avaf..oya Jlf: 'tTJ q>U<Jll tOU npo�A.iJJ.lato�.

II.x crto np6�A.11JlU Jla� 11 tnaA.i]ew<Jll Jlnopti va yivtt J1t to va �pOUJlt avaA.unKa to E.K.II Kat va f.1ttpi]crouf.1t tov apteJ.16 trov noUanA.acrirov trov 1 0, 1 2 Kat 1 5 nou tivat nptv an6 to E.K.II yta va �pOUJlt n6crt� <pop€� txtt tmO"Kt<ptti o Kaeeva� to XOOpt6 Jl€XPt 'tTJV 11Jl€pa nou O"UVavri)e11KaV Kat 01 3 yta npffi'tTJ <popa. llpo�l..fi JlU 2° 'Eva� aveonffiA.11� EX£1 48 K6KKtVa, 40 acrnpa Kat 1 6 Kitptva yapi<paA.a. a) II6crt� to noM OJ.l010Jl0p<pt� aveooecrJlt� JlnO­ pti va <pna�tt J1t auta ta yapi<pnA.a; �) An6 n6cra yapi<paA.a Kaet .XPIDJ.lat� ea anott­ A.titat Kaet aveooBO"Jl11; Bi]Jla rrproto : uvuyvroptO't) otooptvmv, �fiTO'UJlE-

------

V(I)V KU\ (j)pclO't(I)V «Ki..ttOlcl» AtOOJltva: 0 aveorriDA11� EXEt 48 KOKKtVa, 40 U0"1tpa Kat 1 6 Khptva yapi<paf..a .

ZfiTOUJltva: a) rr6crt� to rroM OJ.I.Ot6J.10p<pt� aveooBO"Jlf:� JlnOpti va <ptta�tt; Mt <ppacrtt� «KA.ttota» i) TO no#.\) ii)OJlOlOJlOp(j)t� �) ano rr6cra yapi<paA.a Kaet XPIDJ.lato� ea BXEt Kaet aveoOEO"Jlfl; BiJJla of:Un:po : ANAAYl:H OtOOJlEvOOV, s11tOUJ.lE­ vrov Kat <ppacrtrov «l<Attota».

OJ101.0J10p(j)t� <Jll !laivtt, ot aveooecrJlt� Jlf:ta� tou� va ntpttxow tov ioto apteJ.16 K6KKtva, tov i­ oto acrnpa Kat tOV ioto Kitptva yapi<paA.a. To no#..u <Jll !laivtt tOV Jlt'YnAUttpo apteJ.16 OJ.lOlOJ.lOp<prov aveootO"JliDV nou JlnOpti va <ptta�t1 0 aveoniDA11�· a) Enttoi] txouJ.lt va JlOtpacrouJ.lt ta 48 K6KKtva, ta 40 acrnpa Kat ta 1 6 Khptva yapi<paf..a O"f: OJ.I.Ot6Jl0p<pt� aveoOEO"Jlf:�, Kataf..a�atVOUJ.lf: 6tt 0 apte­ JlO� trov aveootO"JliDV ea Otatpti tOU� apteJ.loU� 48, 40, Kat 1 6. t111A.aoij ea tivat KO\VO� 0\UlpE'Tfl� TOt>�. To np6�A11Jla s11tUtl va �pOUJ.lf: to Jlt'YnAU­ ttpo an6 tou� Kotvou� Otatp€tt�, apa npentt va �pOUJlt tov M.K.A trov apteJ.lroV 48, 40 Kat 1 6. �) E<p6crov ta yapi<paA.a JlOlPUsOVtat O"'tt� aveoo€­ O"Jlf:� J1t tOV JlEy10"t0 KOtV6 OtatpE'tTJ M.K..:1, Otv txouJ.lt napa va otatp€crouJ1t 'tOU� apteJ.lou� 48, 40 Kat 1 6 J1t tOV M.K.A. !:YN9E!:H EllAAH9EY!:H H O"Uvet<Jll Kat 11 tnaA.i]etu<Jll a<pi]vttat crtov ava­

yvfficr'tTJ. llpoTttVOJltvU npo� A.UO't) rrpo�A.'ll JlaTa llpo�AfiJla 1°: 'EXOUJ.lf: 1 26 tttpaota, 1 1 2 JlO­ A.u�ta Kat 42 y6Jlt�. a) II6cra to noM OJ.lotOJ.lOp<pa OBJ.lata JlnopoUJlt va <pna�ouJ.lt; �) II6cra tttpUOta, JlOM�ta Kat y6J1t� ea ntpt€xtt to Kaet o€Jla; np6�A.f1Jl« 2°: H K<wnava Jlta� tKKA.11cria� xmna Kaet 6 A.tnta, JlUX� au11� Kaet 9 A.tnta Kat Jlta� tpi'tTJ� Kaet 1 5 A.tnta. AV xmrri]crow 6A.t� Jla­ si crn� 8 n.Jl, a) Ilota ropa ea �avaxmrri]crow 6A.t� Jlasi yta npro'tTJ <popa; �) II6crt� <pop€� ea EXEt xmrri]crtt 11 Kcl9t KaJ.lnU­ va JlEXPt 'tTJ crttyJ.liJ nou ea �avaxmrri]crow 6A.t� Jla�i yta npffi'tTJ <popa; llpo�l..f1 J1« 3°: .:1uo JlOtocrtl<Atncrt€� avaxro­ pouv an6 tO ioto <Jll fltlO A O"tOV ioto op6JlO, npo� 'tTJV iota Kattuew<Jll . 0 nproto� tp€Xtt Jlf: 60 km/h Kat o of:Uttpo� JlE 80 kmlh Kat �tKtva 1 h apy6tt­ pa an6 tov nproto. �t n6crt� ropt� an6 'tTJ crnyJ.liJ nou �tKiVTJcrt o nproto� Kat crt nota an6crta<Jll an6 to <Jll J.lf:lO A 0 OtUttpo� ea <ptUO"f:1 tOV nproto;

EYKAEIAIII: A' 89 -r.l/15

•.

1. Apxl] an6 n; xropt� T11� avaTOAll�

H AS�11 rt:ropt:Tpia £tVCl1 £AA11V1Kft m>v9s­ Tll an6 � At�e� r11 (fta) Ka1 Mtrpo (Mt­ TP11CJ11 ) . 01 apzaio1 A.aoi sixav TCl xropci<p1Cl TOU� 0'£ O'XTJilCl op9oyroV10 1tA.ci1 0'£ 1tOTClf.lOU�.

IIoA.u O'UXVci :xps1cil;ovTav va f..1.£tpouv TTIV sm­ cpcivsta KCl1 TTIV rrspif.l£tpo tou�, 11 orroia £7t19u­ f.lOUcrav va txs1 rrspirrou icrs� 01acrtcicrs1�, y1a va crto1:xil;s1 <p911v6tspa 11 KataO'Kwl] TTI� rrs­ pi<ppa�� tou� 6tav au� Katacrtps<p6tav arr6 ta rrotclf.l1Cl rrou 1tA11f.lUpli;av. 'Etm zropci<p1 1OOm2 f.l7topoucrs va t:xs1 01acrtcicrs1� 50m Kat 2m KCl1 rrspif.l£tpo 50+50+2+2=104m, av Of.lOO� tO zropcl<p1 �tClV tetpciyrovo £1t£1b� 2 1OOm =10· 10 9a size 1tspif.l£tpo 10·4=40m A1t6 apzat6tspou� :xp6vou� Ot A.aoi crta cr�A.a1a tKavav ropaia crxflf.lata, KCl1 cr� £1t1<pcivste� tOOV Cl'Y'Y£t00V � 0'£ �AtV£� 1tA.ciK£� KCl1 1tci1t1>pou�.

01 BapnA.rovwt a1t6 to 2500 1t.X. sizav ta zropci<p1Cl tOU� 1tAcl1 crtou� 1tOtClf.lOU� Tiyp11 KCl1 EucppciTTI. l:to BpstaV1K6 Moucrsio U1t<XPX£1 f.l1Cl �A1Vll 1tA.ciKa f.l£ 5 KUKAoU�, 1tOU 0'£ Ka9t­ va a1t6 autou� siva1 4 t6�a.

A1t6 to Iln9ay6ptw 0t<f>P11f.lCl ��spav tp1cios� 1tA.wprov op9oyroviou tp1yrovou: (3,4,5), (5,12,13), (8,15,17).

'H�spav aKOf.lCl ta £f.l�aoci Tou tp1yrovou, op9oyroviou, tpa1tsl;iou, cilCA.ou fl£ tov ro1to E = �( OUXJ.1Stpo�)2 61tou to 1t = 3.

4

E1ttCJT)� ��spav Cl1tAoU� 6yKou�. A1t6bst�11 Tll� avci1tro�11� Tll� fsrof..1.£tpia� sivat to 1tp6�A11f.l«. «Na v1roA.oyunei 17 a�aiva rov 7repzye­ ypappevov (Je rpiywvo JdJKA.ov».

O t Atyimnot KataO'Kffiai;av Kat autoi ta xropci<ptCl tOU� 01t(J)� Kat Ot Ba�uA.c0V10t bt1tAa crta uoato<ppciyf.lata tou NsiA.ou, 1tou 1tA11f.lJ.I.U­ pti;s Kci9s :xp6vo. 'Eypacpa.v crs 1tci1t1>pou� a1t6 os�tci 1tpo� ta aptcrtspci. 0 1rawpor; f'f/r; M6axar; Cl1t0 tCl 5 1tpO�A�f.lCltCl 1tOU E:X£1 ta 4 sivat ysrof..1.£tp1Kci. Eyvropti;av tou� ro1tou� trov £f.l­ �aorov tptyrovou, op9oyroviou tpa1tsl;iou, ci­ KA.ou fl£ tU1tO Cl1t0 tOV 01t0t0 E�'YCltV£

1t =

( 9) 2 15

=:

3,1604

Ot Ktvt�ot u1tocrT11pii;ouv 6n sixav f.l£Ae­ �crst 1tptV a1t6 cillou� A.aou�, 61tro� Kat to Ilu9ay6psto 0srop11f.lCl 61tou

P a2 + 4 · y ( r - P) 2 + 4Py 2

·

=:

T� nvti;tK£� ysrof.lstptKs� yvrocrst� sivat 3'60'KoA.o va n� f.l£Ae�crouf..1.£ £1t£tb� tov 3° 1t.X atrova o autoKpcitopa� L1X X1avyK 3ltta�s va Kaouv 6A.a ta b1bClKttKci �t�A.ia. IIoA.M a1t6 autci �avaypci<pTTIKClV xropi� Of.lOO� Cl1t0bst�£t�.

Ot Ktvel;m Kat m Ivooi smcr�f.lOV£� tovii;ouv 1tOCJO 1tClAtcl sivat 11 f£00f.l£tpia tOU� f.l£ tO «1£­ po Bt�A.io» Kat to 1toi11f.l« «l:ouA.�acroutpa» trov Ivorov 1tou fl£ to Bou3tcrf.l6 ypcicps1 yta TTIV KCltClO'lC£1>� tstpayrovrov Kat op9oyrovirov, CJXS­ CJ£1� 3tayroviou Kat 1tA.wpt� tstpayrovou, Ka90>� Kat tstpayrovtcrf.l6 cidou.

LTTIV avci1tro�11 Tll� rsro!l£tpia� £Kt6� a1t6 tOU� 'Ell11V£� 1tOU �tav 1tprot01tOpOt O'UVE�Cl­ A.av 01 IItpat�, m BapnA.rovtot m Jvc)oi Kat m 1\paPt�.

EYKAEIAm A' 89 T.l/16

AuvovTac; 1t p o pA.ft J!aTa J!E E�taroaEtc; Kat ypa<ptKtc; 1tap aaTUO'Etc; EU 9EtroV =======

Tot>pvaiJinJ<; �Ttpyw<;

IIoA.Mi 1tpo�A.it J.Ul'ta 'tTJ<; Ka8ru.teptviJ<; l;coit<;, 'tTJ<; 1tpaKnKi)<; apt8f.1TtTtKft<;, <pumKT)<; K.a.

A.uvovrat av «J.lE'ta<ppacrtouv» a1t6 'tTJV <pumKT) !la<; yM:Oc:ma, cr'tTJ yM:Ocrcra 'tTJ<; ciAye�pa<;. Mia e�icrco011 cr'tTJV ciAye�pa, eivm 'to tcroouvaf.1o flta<; 1tp6'ta011 <; 'tou 1tpo�A.itf.1a'to<; it oA6dTtpou 'tou 1tpO�Aitf.1a'tO<;. IltptiX,tt 'tOV ayvcocr'tO (cruvfJ8co<; crl>f.1�0Ati;t'tat J.J£ X) 'tOt><; yvcocr'tOU<; apt8f.10U<; 1t0l> cruvoeovrat J.lE'ta� 'tOt><; J.l£ n<; 'tEcrcrtpt<; 1tp�tt<; Kat flta tcrO'tTJ'ta.

H tmAUOTt flta<; e�icrco011 <; �acril;e'tat O'tTJV tKf.1c18TtOTt Kcl1totcov cruyKtKptf.lEvcov 'ttX,VtKffiv Kat Ka'taA.ityet (Ka'ta 'tTJ yvffif.1Tt f.1a<;) cre eva a1tA6 sit'tTJfla. To mo oucrKoA.o eivm va Ka'tacr'tpfficrouJ.J£ 'tTJV e�icrco011 . A<; oouJ.J£ J.J£ptKa 1tpo�A.itfla'ta 1tou A.uvovrat J.l£ 'tTJV �oit8tta 'tTJ<; J.lE'ta<ppaOTt<; cr'tTJ yM:Ocrcra 'tTJ<; ciAye�pa<; Kat 'tCOV e�tcrfficrecov 1tou 'ta tK<ppai;ouv.

1.

EKi'lfOV evav apl()po, rpl1r).auiaue rov, 1rpou8eue 7 Kal Oa ppel� 19. Ilow� eivaz o apl()po�; Atl(jll :

A ' Tporro� : 1

ll iv a. Ka

a�:

X

3x

3x + 7

7

3x + 7

= 19

rta va 1tpocrotopicrouJ.J£ TOV apt8f.16 apKei va ).6uovue 'tTJV TeAeu'taia E�icrco011 .

2

6 cov otatpouJ.J£ J.l£ Tov cruvreA.tcr'ti) 'tou ayvfficr'tou

3x =

12

3x

12

x = 4 E rr cJ.i,q 0£u(jlJ : 0 apt8f.16<; 4 J.l£'tacrxlJf.1atil;t'tat Kat Ka'taA.ityet cr'tov cr'tftA.Tt 'tOt> 1tivaKa 1 .

1 9,

crUf.1<pcova Kat J.l£ 'tTJV 2"�

4

3 · 4

3 · 4 + 7

19

B 'Tporro�: A1t6 'tTJV e�icrco011 3x + 7 = 19 av a<patpecrouJ.l£ 'tov 19 EX,OUJ.l£: 3x + 7 - 1 9 = 19 - 19 it 3x - 12 = 0.

EYKAEIAID; A' 89 T.l/17

------

A'6vovr� npoiJA.qpaTa J1t: �tCJ6>crt:� Kat ypu4ptK� napaGTaCJt:� t:1>9t:t<Ov

-----­

Ms 'tTIV poi)esta 'tou 3u!OtKroou Ka'tspa�OUJJ£ Kat syt<aetcr'tOUJ.lS 'tO "Geogebra". �'tll cruvtxsta 1tAl'JK'tpoA.oyo'6JJ£ y = 3x - 1 2 cr'to 1ts3io 'tll � stcrayOlY'l� (Ka'tco aptcr'tspa) Kat av TO KclVOUJJ£ O'COO''tU, ea 1tUpOUJJ£ 'tTIV Eu8tla 3 •: v • 3x t2 3mA.avi) stK6va.

2 llapa'tll p0'6JJ£ 6n o apteJ.L6� 4 1tou ppi)KaJ.Ls 1tpolJyouJ.Ltyco� co� A.'60'l'J , sivat &Kef 1fOV -ripV8l '1 ev(Jefa 1f0V 1rpOBKVf/18 9 -:: 7---:8c-6 ---=- 1 -"t:0:---!1�2::---o _3 _2::-_c l---': 3 --t�5r--" 1 pe-r a 'r't/ V eu;a'YOJ'Yt/ 'rOV nJ1fOV y = 3x - 12, TOV Opl(OV'rlO a�ova x 'x.

-2 fta va 'tO S�l'JyTJO'OUJ.lS aU'tO -3 XPSta�6J.Lacr'ts A.iya A.9yta a1t6 'tTl escopia 'tOU S1tOJ.lSVOU KS<paMtOU 'tCOV cruvap'ti)crscov.

A� �SKtvTJO'OUJ.lS a1t6 J.Lta 1tapa'ti)plJO'l'J. Ot S�tO'IDO'S� 1t01) ea J.La� a1tacrxoA.i)crouv O"tTTV B 'tcl�l'J sivat a' paeJ.Lo'6.

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TTt cru<JXEtl<JTt TOU TtAtuTaiou tpro'Ti]J.laTO<; J.1t Ta OUO 1tpOTt'YOUJ.ttVa, a<pi]VOUJ.lt va TTJV a1tOKMU'Jittt J.lOVOt cra<;.

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Eppaoa Kat. n\)Oayoptt.o 9t<i>Ptt J1«

To ABr AEZ CJTI)V 1tUj)U1tclV(l) £t.KOVU dvut.

Ev« KUVOVt.KO t;cl"(IDVO. (01..£c; Ot. 1tM:\lpic; d­ V«t. iatc; K«t. ol..£c; Ot. yiDvi£c; tiV«t. iatc;.) AV AB=4 cm, p1topti1'£ vu q»tp&Tt tatic; Kcl1tOt.« Pott&qTt.Kl} Et>9du cOGT£ vu \)1tOAoyiG&T£ TO .:ppuMv 1'0\) ABrAEZ; Av q»tp&T£ ol..tc; 1't.c; Ot.«ycbvt.o\)c; U1t0 1'1)V KO­ j)\)(J)l} A p1topd1't vu \)1tol.oyia&Tt T« .:ppuoa 1'IDV Tpt.ycbviDv 1t0\) 9« <JXTIJ1«1't.G1'o1>v; Nu l)1tOMyiG&T£ 1'0 tppuoo G't'U TpiyiDV« 1t01> 1>1tclf>X01>V «j)K&Tcl G1'0t.X£i«.

3.

E r

DY0ArOPEIO 0E!!PHMA :

I:t Kcl9£ op9oycOVt.O 1'piyiDVO 1'0 1'&TpayiDVO 1'1)� 1>1t01'£iVO'UG«c; dV«t. {GO p£ 1'0 a9pOt.GJ1U 1'IDV 1'&Tp«ycOVIDV 1'IDV 01>0 Kcl9&TIDV 1ti..£'U prov.

«2 = P2 + l

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AiV&TUt. 1'0 1tUj)«Kcl1'(l) axl}pu:

A

B

Tu TpiyiDv« ABE, BrE, rAE dvut. op9oycbvt.u.

Av EA=3 cm K«t. EB=5cm K«t. AB=Br=rA vu 'U1tOAoyiG&T£ ouc; 1't.c; 1tM:\lptc; 1'01> axl}p«1'0�.

4.

To T&TpayiDvo ABrA fxtt. 1tl..t'U p a u=13 cm.

Av 1'0 op9oycbvt.o TpiyiDvo AAB fxtt. AA=5 cm vu 1>1toloyiat1'£ To .:ppuoov 1'01> T&Tpuycbvo-u uv «q»«t.pt9£i 1'0 1'piyiDVO. � -------. r

To eeropru.ta a\)'tQ 'tO typa'JIE 0 aPXaio<; 'EA.A:rtva<; q>tA.ocro<po<; Kat Ma9ru.tanK6<; Ilu9ay6pa<; Kat mtaPX£t mo 1tpro'to PtPA.io <na 'I:Tmxeia' 'tOU Eu­ KA.£\511 f!a�i f.l£ O"X£'tUCft a1tOO£t�ll · I<JXt}tt. K«l 1'0 «VTiGTpoq»o 1'01> II-u9uyopdo-u 9tiDpl}p«Toc;: Av cre tva Tpiyrovo TO 't£'tpayrovo 'tll <; f.1.£yaA.U­ 't£Pll <; 1tAzupa<; eivat icro f.1.£ 'tO a9pOtcrf.1.a 'tiDV 't£­ 'tpayrovrov Trov Mo MA.rov 1tA.f:uprov TOT£ TO Tpiyro­ vo eivat op9oyrovto f.1.£ op9ft 'tl1 yrovia a1ttvavn a1to 'tl1 f.l£'YaAU't£Pl1 1tA.cupa.

E<papf!O'Yi� 1.

To TpiyiDvo op9oycbvt.o ABr fxtt.

A = 90°

K«t. B = f' = 45° . Av AB=l dm vu 1>1tOAoyiG&T£ Tt.c; 1>1tolomtc; 1ti..£'U ptc;.

2.

I:To 1t«p«KU1'ID axl}p« To ABrA dvut. op9oycbvt.o pt 1tl..t'U pt� AB=lOcm K«t. Br=4cm.Av AM=rN=3cm vu 1>1tol.oyia&T£ Tt.c; 1tM:ll pt� Too Tp«1tt�ioo ABNM K«9roc; K«t. To .:ppuoov To-u.

5.

To XIDpaq»t. 1'01> 1tauo1> fxtt. axl}p« t.aoaK£­ w-.>c; 1'j)U1tt�t.O'U. 0 ol}poc; UVUKOiVIDG£ 01't. U1t0 1'0 XIDPcl(J)t. 9« 1tpi1ttt. V« 1t£j)UG£t. 0 vtoc; opopoc;. To .:ppuoov To-u XIDP«q»t.o-.> l}T«v 1 GTptpp«. To 1'pl}p« 1'01> opopo'U 1t01> 9« 1t£j)cl(J£t. pia« U1t0 1'0 XIDPcl(J)t. 9« fxtt. 1tMIToc; 5 m all« o 1tauo-.>c; otv aKO'UG£ Tt. pl}Koc; 9« fxtt..

I.

II.

Av dvut. 01tiDc; GTO axl}p« TO XIDpaq»t. pt TO Oj)O JIO J11tOj)£i1'£ VU pp£i1'£ 1'0 pl}Koc; 1'01> opopo-u; BK=40m EK=14m,EZ=5m.

M1topd1'£ vu 1>1toloyia&T£ 1toao 9« yivtt. pt1'cl 1'0 opopo 1'0 tpP«Mv 1'01> XIDP«q»to-.>;

B

M

E

N

EYKAEIAHI: A' 89 T.l/23

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nAPAro Nro n o i H l: H

Ko •

TAYTOTHTE:r

01 1'Ut>1'0TI11'E<; tiV«l tv a UJXt>PO tpyalf:io 'Yl« TllV napayoV1'01t0tfl<fl) 1'0>V aA.ytpplKcOV napaO't'UCJf:O>V.

Ta�fl

======

D.

Xpunono1>J..o� - 9. Xpunono1>J..o�

A. TtA.ew. TtTpaywva

(a + p)2 = a2 + 2ap + p 2 (TtTpaycovo a9poicrJ.taT�) (a - p)2 = a2 - 2ap + p 2 (TtTpaycovo ota<popa�)

B. Tt A.e w t

K i> f3 o t

(a + P)3=a3 + 3a2P + 3aP2 + P3 (ri> Po� a9poicrJ.taTo�) (a - p)3=a3 - 3a2P + 3aP2 - P3 (ri> Po� ota<popa�)

C. A.Opo t(jJta

erri () w <p o p a

(a + p)(a - P)=a2 - P 2 (ota<popa TtTpayrovcov)

D. AKOJ.ll1 1ttpt(j(j0Tf:ptc; T«lUTOTllTES . . . .

(a + P + y)2=a2 + P2 + y2 + 2a P + 2ay + 2 PY (a - P + y)2 =a2 + P2 + y2 - 2aP + 2ay - 2py (To 1tP00'11 JlO O'Ta Ot1tMicrta ytv6J.t£Va 1tpori>1tTtt a1t6 TO ytvOfl£\'0 TCOV 1tP00'11 JlCOV 1t01> EXOUV ot 6p01 O'TO ri>pto 1t00'0 TOU Ka9t JlOVCOWJ.lOU) a3 + P3 + y3 - 3apy= (a + P + r) [ (a - p)2 +(p - y)2 +(y - a)2 ]

�

OJ.lCO� av a + P + y = 0 TOTt a3 + P3 + y3 = 3a py (a + P)2 - 4aP=(a - P)2 Kat (a - P)2 + 4aP = (a + P)2 (x + a)(x + P) =,:e + (a + P)x + aP Kat (a2 + P2)(x 2 + y2)=(ax + Py)2 +(ay - px)2

I:xoA.w : llapairfp�me arz ae 6k� rz� 1Capa1Cavm ravroirfre� ro iva a1C6 ra f56o f.JBA:'l eivaz yzvoJ.Jevo.

H A PAfONTO O O I H I: H

•

11apayOVT01t0l110'11=(1totc0 1tapayoVT� OllAOOll qm{xxvco 1tapayovt� apa JltTaTpEnco Eva a9pot­ O'J.la i] Jlta otaq>Opa cre ytv6J.ttVo i] a/J.� Jlta 1tp6cr9t0'11 i] a<paipeO'll cre 1toAA.rutA.acrtacrfl6) ftan; Tt XPtta�tTat 11 1tapayoVTorroi110'11 ; Ma9aJJ£ 1r.x. on: (x - 2i = x 2 - 4x + 4 . �e Ka1tott� 1ttpt1tTrocre� Jl� t9>1t11 PtTei 6Tav sxouJ.tt TO 1tpc0TO ,..EA.o� va ypa'lfOUJ.lt TO OeUTtpo Kat O't WJ...t� TO avricrTpO<pO. A� OoUJ.lt Tffipa O't 1t0ffi� 1iept1tTrocre� Jla� t�1>1t11 PtTti KaA.:6Ttpa 11 1tapacrTaO'll (x - 2)2 a1t6 'tllV x 2 - 4x + 4 .

I i . Tt eivat euKOAOTtpo va A.:ucrouJ.tt 'tllV e�icrcoO'll x 2 - 4x + 4 0 i] 'tllV (x - 2)2 = 0 ; Ilpo<pavro� 'tllV (x - 2)2 = 0 ot6n x - 2 0 OllAOOll x = 2 x 2 - 4x + 4 11, TO ( x - 2)2 • , , 1 1 . Tt tlVat et>KOJWTtpO Va a1tJW1t011lO'OUJ.lt (av X :t; 2 ) TO IVWO'Jla x-2 , x-2 Acr<paA.ro� TO OeUTtpo. i i i . Tt eivat et>KOAOTtpo va ppouJ.tt TO 1tP00'11 JlO Tll� x 2 - 4x + 4 i] Tll� (x - 2)2 6Tav x= l i] 2 i] 3 i] 2,0 5) ; Ilpo<pavro� Tll� (x - 2)2 . i v . E1ti0'11 � 9a XPttacrTOUJ.lt 'tllV 1tapayoVT01tOtl10'11 yta 'tllV rnptO'll TOU EKTI Kat TOU MU 1ta­ pacrTacrecov. fta 'tllV 1tapayoVT01tOl110'11 9a XPtta(jTOUJ.lt � TaUTO'tllTt� a1t6 'to OeUTtpo ,..EA.o� 1rpo� TO 1tpcOTO, 011A.a.oi] a1t6 'tllV av01ypivtJ O''tllV napayovronol'IJlivtt JlOp<pi].

•

=

=

'l A

"� -

.A'I .!.

EYKAEIAm A ' 89 ,;.1/24

-------

llAPArONTODOI:tn;H K«I TAYTOTHTEI:

------

n w� Y lVtTH l 11 1t'Up uyovT01t'O l ll (j11 ;

B r uw 1 o : E�s-ra�OUJ.LS xpoasxnKa uv ot ouo opot txouv Kotvo xupayovra (xpoaoxfl ! Kat axo -rou� apd}JlOU� Kat axo -ra «ypaJlJla-ra»). Av vat, ypa<pOUJ.lS -rov Kotvo xapayovta Kat avoiyouJ.lS xaptvOsCfll J.ttaa a'fllV oxoia ypacpouJ.lS «on axoJ.ttvst» axo -rou� ouo opou�. (OTJMxOit -ro 1tTJAtKo KaOs opou ota -rou Kotvou xapayovra) n upa()unw : 6a2� - 4 a2�2y = 2a2 �(3 - 2�y) .

I:zol.w t 0 : Av tv� opo� �aivst oA.OdTJpO� aav Kotvo�, -ro-rs a'fll O£Cfll -rou J.ttvst JlOVaoa. fta xapaostyJla a� + a = a(� + 1 ) . l:zoho 2 ° : Av OOUJ.LS ouo avtiOs-rs� xapaa-raast�, 61tro� a - � Kat � - a -ro-rs apKsi va OtaouJ.lS Jlta axo � ouo as Jlta xaptvOsCfll Kat Jl1tpoa-ra -ro xpOCfll JlO (- ), £-rm Oa ooi>J.LS on txouJ.lS Kotvo xapayov-ra x.x. (a - �)= - (a - �) . AKOJlTJ as xapaa-raas� xou sivat mvroJ.ttvs� as �uyo sKOE'fll

Jl1tOpol>J.LS va aA.Mi�OUJ.lS -ra xpOCfll Jlcl -rou� 1t. X· (� - a) 2 =(a - �)2 • H n p a()unw :

a(x - 2y) + �(2y - x) - (2y - x) 2 = a(x - 2y) - �(x - 2y) - (x - 2y/ = [ ( x - 2 y)(a - � - (x- 2 y) ] .

B !lJ.w

t• : E�s-ra�OUJ.LS av ot 2 opot sivat it Jl1tOpouv va yivouv otacpopa -rs-rpayrovrov 8TJA<lOit av txouJ.lS 'fllV -rauTO'fll Ta: a2 - �2 yivs-rat (a + �)(a - �) . fl u p a 6 ny !Ja I u :

4x 2 - (x - 3) 2 = (2x) 2 - (x - 3) 2 = [ 2x + (x - 3) ][ 2x - (x - 3) ] =

=(2x + x - 3)(2x - x + 3) = (3x - 3)(x + 3) = (x - 1)3(x + 3) = 3(x - 1 )(x+ 3).

11 a p 6()uy f! u 2 " : :EzoJ.. w 3" :

20x - 45xy 2 = 5x(4 - 9y 2 ) = 5x [ 2 2 - (3y) 2 ] = 5x(2 + 3y)(2 - 3y).

Av ooi>J.LS xap£V0£ast� Kat nxo-ra axo -ra xapaxavro OEV yivs-rat , -ro-rs «avoiyouJ.lS» 'fllV xapa<JTaCfll Kat aPX�OUJ.lS axo 'fllV apxfl txov-r� xtOavov xsptaao-rspou� opou�. 8TJMxOit o­ -rav OOUJ.LS «Jltao-xapayovt01tOtTJJ.t£v'TJ» xapaa'taCf11 uxapxouv ouo £VosxoJ.l£Va, it va <JUvsxiaouJ.lS xavro a'tllv iota JlOpcpit it au'tft va JlTJV oOTJysi as xapayovroxoiTJ<rll oxo-rs «yupi�OUJ.LS maro» J.lS clVOtyJla 't(l)V 1tap£V0EaSOOV. ll u p a o ny!Jn l : a(x - 2�) + x 2 - 4� 2 =a(x - 2 �)+(x + 2�)(x - 2�)=(x - 2�)(a + x + 2�). Eoro, 11 xa­ ptvOsCfll �OTJ0ouas va <JUvsxiaOUJ.lS. n u p {.tottyJ.1a2 : a(x - 2 �) + 3a� = ax - 2 a� + 3a� = ax + a� = a(x + �). Eoro TJ xaptvOsCfll OEV oOTJyouas as xapayovtoxoiTJCfll . ea <Jl>Vav'tftas-rs xoW� xapaa-raast� j.ts ouo opou� 1t0l> 0£\1 1tapayoV't01totOUvtat XX· X 2 + 1 ' x 2 + y2 (ysvtKcl 'tO aOpotaJla 'tS'tpayrovrov it ysvtKcl 'tO aOpotaJla OsnKrov apt9J.1IDV) 0 a p a c'k ty JlUTU i) ii) i ii)

iv)

x 2 - xy = x (x - y) .

3a� 2 - 3 a = 3 a(�2 - 1 )=3a(� + 1 )( � - 1 ) 2x 3 y - 8 xy3 = 2xy(x 2 - 4y 2 ) = 2xy [ x 2 - (2y) 2 ] = 2xy(x + 2y)(x - 2y).

5(a + 2�)2 - 2 0 = s [ (a + 2�) 2 - 2 2 ] = 5(a + 2� + 2)(a + 2� - 2) .

(2x - 3y) 2 ..,- (x + y) 2 = (2x - 3y + x + y)(2x - 3y - x - y) = (3x - 2 y)(x - 4 y). v) Biuw 3 11 : Av sivat 4 ot opot s�s-ra�OUJ.LS av sivat it Jl1topouv va yivouv TtM:toc; .ropoc;.

8TJMxOit av txouJ.LS: a 3 + 3a 2� + 3a�2 + �3 TOT£ yivs-rat (a+�)3 it av txouJ.lS a3-3a2 � +3a�2 -�3 01to-rs yivs-rat ( a-�)3 • 0 u p a i> £lYf.W 5K3 - 30�+ 60K-40 =5(K3-6�+1 2K-8)= 5(K3-3 · 2K2+3 · 22K-2 3)=5(K-2)3 BfJ J.Hl 4" : Xropi�ouJ.lS as 2 OJla�s� -rrov 2 oprov Ka9E Jlta Kat �ya�OUJ.LS Kotvo xapayov-ra Ka-ra o­ Jlcl.Oa, J.lS a-roxo ot ouo xap£V9£aEt� xou 9a xpociwouv va Eivat iote�. Av OEV Eivat io� TOTE 00KtJla�ro {i)J..o <JUVO'OaaJlO Kat av a1tOTUXOUJ.lS Kat J.lS 'tOl>� 'tpEt� <JUVOUaaJlOU� 'tOTE E�E'tcl�Ol>J.LS 'tO 5 ° �itJla. TiapaOEtyJla 3 K3 + 6K-�A-2A = 3K(�+2) - A.(�+2)= (K2+2) (3K-A) .

EYKAEIAID; A' 89 T.l/25

------ DAPAI'ONTODOIHI:H

K«l

TAYTOTHTEI:

....;__ _ _ _ _ _ _

_ _ _

Bqpa 5°: E�s't6.�oUJ..I£ av ot 'tpst� 6pot a1tO'tsA.ouv 'ttM:to 'tE'tp6.yrovo Kat S11J.Ltoupyouv J..l£ 'tOV 'tt­ 'tapto Stacpop6. 'tE'tpayrovrov. llapaostypa 1o : a2+2 aP+ P2- i = (a+Pi- y2 =(a+j3-y)(a+ p+y) 2°: x2-\li+2'JI- 1 sSro Ssv u1t6.pxst Kotv6� 1tap6.yrov, Ssv sivat 'ttM:to� ci po� Kat K6.6s (J'UV­ SuacrJ.L6� xroptcrJ.LOU crs OJ.L6.Ss Ssv Sivst iSta 1taptv6s<J11 , 01tO'tS S�S'tU�O'UJ..I£ 'tO 5° PilJ.LU

1

•

x2-{'JI2-2'1'+ 1 )= x -{'JI- 1 i= [x-('JI- 1 )][x+('JI-1 )]=(x-'JI+ 1 )(x+'JI-1) Ilap6J.Lota <JlCE1t'tOJ.LUcr'ts Kat 6'tav tXO'UJ..I£ 1tEptcrcr6'tspou� ( 6 i] 8) 6pou�.

A(JK..,(J£1�

Na yivouv ytv6J.LSVa:

20K2+60K+45

2. a2-3aP+2P2 3. x2-7x+ 10 4. a(p2-y2)+P(y2-a2)+y( a2-P2)

Au(J11

1.

20K2+60K+45 = 5(4K2+ 12K+9)=5(2K+3i

4.

(x-y)(x+3y) x2-3 y2+2xy=(3°� 'tp61to�, 'tptOOV'U flO ro� 1tpo� x) = x2+2y · x-3l (..1=16l , X1,2=y i] -3 y) o1tO'tE x2- +2xy=(x-y)(x+3y) a(p2-y2j+ p( -a2)+ y(a2-p2)=ap2-ar+ py2-pa2+ �a2-yp2 =\1tpocr6t'tro .Kat acpatpro 'to apy) = ap2-ay + P -Pa2+ ya2-yp2+apy-apy =aP2-apy-p y+Pr2- a P+a2y+aPr-ai = =aP(f3-'r)--l}y(j3-y)-a2 (f3-'y)+ay(j3-y)=(13---y)(afH3y-a2+ay)=(f3-'y)[p(a-y)-a(a-y)]= (f3-'y)(a-y)(13--a) .

2. a2-3 aP+2P2=(St6.crxa<Jll 6pou)= a2-2 ap-ap+2P2 =a( a-2P}-P(a-2P)=(a-2P)(a-p) 3. x2-7x+ 1 O=(J.Ls St6.cr1tU<Jll J..lCcraiou 6pou)= x2-2x-5x + 1 0= x(x-2}-5(x -2)= (x-2)(x-5) Kat cpumK6. 'ti1to'ta Ssv Eivat J.Lov6SpoJ.Lo� . . . va 'tps� StacpopsnKoi 'tp61tot yta va 1tapayoV'to1toti]crou J..l£ 'tllV iSta 1tap6.cr'ta<Jll : • x2-3l+2xy=(1°� 'tp61to�) = x2-l-2l+2xy=(x-y)(x+y)+2y(x-y)=(x-y)(x+3 y) • x2-3l+2xy=(2°� 'tp61to�) = x2+2xy+l--4l=(x+yi-(2yi=(x+y-2y)(x+y+2y)=

�

•

EcpapJ.Lort� 1. rta 'tU 'tE'tp6.yrova 'tffiV <p'UcrtKOOV apt6J.LOOV 1t.X.

pt6J.L6� sivat 'tO 422 =

;.

1 1 , 25 SXO'Uf.!S: 1 12 = 121

i]

252 = 625, 1t0t0� a­

AU(Jl]

apt6J.L6 J..l£ 't1l J.LOpcpi] (1 0·4+2)2 'tO'tE tXO'UJ..I£ 102.42+2. 10.4.2+22=1 600+ 160+4=1764. Tropa J.L1topoUJ..I£ va 1tOUJ..I£ J.Lta f.lS6oSo yta 'tO 'tE'tp6.yrovo K6.6e St'Jiil cptou 1tou A.i]yst crs 5 S11A.aSi] av X 'tO 'JI11<pio 'tffiV SsK6.Srov txouf.!S ( 1 OX +5)2= 102• X2+2. 10. X.5+52 i] (lOX +5i=IOOX2+100X+25=100 X(X+1)+25 o apt6J.L6� 6J.Lro� 100 X(X+1 ) A.i]yst cre Suo J.L11SSVtK6. S'tcrt av 1tpocr'te6Ei 'tO 25 o apt6J.L6� 6a A.i]yst cre 25.

Apa 'tO 852 = sxst 4 '1'11<f>lU 'tU (8.9)=72 Kat UKOAo'U6si 25 S11A.aSi] 7225.

2. Na a1toSe�E'tE 6n 'to ytv6J.LSVO 'tEcrcr6.prov StaSoxtKrov cpumKrov apt6J.Lrov au�11J.L&vo Ka't6. flO­ v6.Sa eivat 'tE'tp6.yrovo cpucrtKou apt6J.Lou. Av

1tapacr'ti]crouJ.LE

'tOV

AU(Jl]

na 1tap6.SEtYJ.LU ot apt6J.Loi 2,

121=1 e.

3, 4 Kat 5 txouv "(lVOJ.LSVO: 2·3·4·5=120 (J'\)V 1

6.pa

121

S11A.aSi]

'Ecr'tro x, x +

3.

1 , x + 2, x + 3 ot 'tScrcrept� StaSoxtKoi cpumKoi apt6J.Loi 'tO'tE: + 1 + 2)(x+3) x(x )(x + 1 =x(x2 + 3x + 2)(x + 3 ) + 1 = [x(x2 + 3x) + 2x](x + 3 ) + 1 = [x2(x + 3) + 2x](x + 3) + 1 =[x (x + 3)]2+ 2x(x + 3) + 1 = [x (x + 3) +1]2 Av v cpumK6� apt6J.L6� 'tO 'tpiyrovo f.1S J.lll1C11 1tAEuprov 2v + 1 , 2v(v + 1), v2 + (v + 1i eivat op6oyrovto; Au(J11

E�e't6.�o'UJ..I£ 1tota a1t6 ne; 'tpetc; 1tM:'Ups� sivat 11 flS'YaAU'tEf11 · 'ExouJ..I£ :

.;+(v+1i=2v2+2v+1>2v2+2v=2v(v+1) Kat v2 + (v +1) =2� + 2v + 1 > 2v +1 6.pa 11 J.LE"faAU'tEP11 1tM:'Up6. eivat 11 v2+(v+ 1 f I:uYKpivouf.!S 'tO 'tE'tp6.yrovo 'tll � J..l£"faAU'tEP11� 1tAEu­

p6.c; J..1£ 'tO 6.6potcrJ.La 'trov 'tE'tPU rovrov 'trov Suo 6.Mffiv : 'ExouJ..I£ [v2+(v+ 1i ] 2=[2vJ+(2v+ 1)]2=4v4+2·2v2.(2v+ 1)+(2v+ 1i= [2v(v+ 1)]2+(2v+ 1i IIu6a­ y6peto 0EOOP11J.LU. EYKAEIAm A' 89 T.l/26

noT[ liuo T p l ywvo [tVO I

loo

Ta 1tAUKCUCta O''tO apyacr'tiJptO Xru.taiac; 'tOU crxoA.Eiou J..lU<; EXOUV crxflJ..la 'tptyrovou 01tffi<; <paiVE'tE crnt Ot1tA.avf) EtK6va.

'Eva a1t6 61troc; �AE1tE'tE EXEt cr1tcicrat Kat eEAoUJ..lE va 'tO avnKa'tacrTfJcroUJ..lE . M1topaha va O'KE<p'tEi'TE n J..lE'tpi}crEt<; 1tpE1tEt va KWOUJ.I.E rocr'tE a1t6 'tO 'tE'tpciyrovo 1tAaKciKt 1tou EXOUJ..lE ayopcicrat va <pnci�OUJ.I.E Eva 'tpiyrovo 1tou va 'tatptal;at aKpt�ro<; O''tO KEVO 1tOU EXEt 011J..lt0UJYY11 eai; Au't6 EtVat 'tO EPOOntJ..lU 1tOU ea J..lU<; a1tacrxoA.i}crat cr'to cru-yKEKptJ.I.Evo cipepo. Tiro<; 011Aa0i} J..l1tOpo'6J..lE va Ka'taO'KwcicrouJ..lE ouo icra 'tpiyrova t1 Ota<poputKci 1tO'tE ouo 'tpiyrova aivat icra. Kci1tmoc; 1tp6'tEtvE va J..lE'tpi}crouJ..lE ne; 1tAwpE<; 'tou 'tptyrovou Kat va Ka'tacrKwcicrouJ..lE Eva 'tpiyrovo J..lE icrac; 1tAwpE<; J..lE am6. IlpciyJ..lan �EKtvf)craJ..lE ntv 1tpocr1tcieata aAA.ci aiOaJ..lE 6n oav J..l1topol>J..lE va 't01toeaTfJcrouJ..lE ne; 1tAEUpE<; roxaia 61tro<; va'vat. Kcin aKOJ..la E1tpE1tE va O'KE<p'toUJ..lE .

11pE1tEt Et1tE Eva<; clMO<; VU J..lE'tpi}crOUJ..lE Kat 1t00'0 ea mp{�OUJ.I.E Kaea <popa, 011AU0i} va J..lE'tpi}crouJ..lE ne; yroviac; -rou 'tptyrovou Kat avciJ..lacra cra ouo icrac; 1tAwpE<; va EXOUJ.I.E icrac; yroviEc;.

Kat va 1tou -raA.tKci -ra Ka'ta<pEpaJ..lE va <pnci�OUJ..lE -ro 1tAaKciKt 1tou eEA.aJ..lE . Ka-raA.a�aivouJ..lE A.m1t6v 6n: Av ol)o 'Tpiyrova txo1>v 'T� 1tU'Uptc; 'T01><; ia.:c; fll« 1tpoc; fll« Kat 'T� aVTiG'TOlX£<; yrovi£c;1 'T01><; ia.:c; 'TO'T£ £iva1 iaa.

IlotO 0'11 J..lUVnK6 yta EJ..lci<; OJ..liD<;, a1t6 ntV 1tP011'YOUJ..lEV11 1tp6-ra0'11 , aivat -ro 6n tcrxl>at 11 UV'ttO''tpo<pi} 'tTl <;: Av ol)o 'Tpiyrova £iva1 iaa, 'TO'T£ 9a txo1>v 'Tl<; 1tUl>ptc; 'T01><; Kat 'T� aV'TiG'TOlXE<; yrovi.:c; 'T01><; ia.:c; f1ia 1tpoc; flia.

Eivat mo O'll J..lavnKi} yta ouo A.Oyouc;: 1 ) ea J..lcleOUJ..lE 'tU Kpt'tiJpta tO'O'tll 'tU<; 'tptyrovrov 'tU 01tOta EtVUt 1tpO'tclO'Et<; 1tOU J..lU<; a�acr<paA.ii;ouv 6n apKouv 'tpia KU'taMT)Aa icra cipta cr-rmxaia ( 1tAwpE<;, yroviE<;) yta va aivat ouo -rpiyrova icra Kt oxt E�t 1tOU 1tpOKU1t'tOUV U1t0 ntV 1tpO'tUO'll yta ntV tO'Ont'ta 'tptyrovrov 1tOU ava<pEpUJ..lE . 'E-rcrt yta va U1t00Et�OUJ.I.E O'tt ouo -rpiyrova EtVat icra ea XP110'tJ..l01tOtOUJ..lE 'tU Kpt'tiJpta tcr6nt-rac; -rptyrovrov.

2) Xp110'tJ..l01tOtOUJ..lE ntV tO'Ont'ta 'tptyrovrov ytati 1tOAU cruxva. J..la<; EVOta<pEpEt va U1t00Et�OUJ.I.E OXt 'tOO'O on OUO 'tptyrova EtVat {cra aMa on OUO wei>ypUJ..lJ..la 'tJ..ltl J..lU'tU EtVat {cra tl On OUO yroviE<; EtVat icrac;. ea J..lE pro'tiJcrE'tE, Kat J..lE 'tO OtKtO crac;, ytati va 'tO KclVOUJ..lE am6; rta VU cruyKptVOUJ..lE U1tEUeaia<; EUei>ypUJ..lJ..lU 'tJ..ltl J..lU'tU tl yrovffi<; J..l1tOpOUJ..lE Ei'fE VU J..lE'tU<pEpOUJ..lE -ro Eva 1tcivro cr-ro ciAA.o, Kcin 1tou OEV aivat Kae6A.ou 1tpaKnK6 t1 aivat aKOJ..la Kat aouva-ro, Ei'TE va 'tU J..lE'tpi}crOUJ..lE, 1tOU EJ..l1tEPtEXEt 'tOV KtVOUVO va KclVOUJ..lE Kcl1tOta EO''t(l) Kat J..ltKpci Acie11 crn<; J..lE'tpi}crEt<; tl ll eEO'll 'tOU<; va EtVat 'tE'tOtU 1tOU VU J..lllV J..l1tOpOUJ..lE VU 'tU J..lE'tpi}crOUJ..lE .

Xpata/;6J..lacr-ra A.m1t6v yta va -ra cruyKpivouJ..lE J..lta a1tOOEtKnKi} J.�.Eeooo, 011A.aoi} J..lta cratpci U1t0 AoytKOU<; cruUoytO'J..lOU<; 1tOU ea U1tOOEUCvUOUV OAO<pcivapa ntV tO'O'tll'tcl 'tOU<;.

'E'TGl o't'av 9tl..o'Uf1E va o.:i;01>f1E O'Tl ol)o £1>00ypaf1f1« 'Tfl'l]fla'Ta 1] ol)o yroviEc; Eiva1 iaa mlltyo'Ufl£ ol)o Ka'TaiJ.:qMI Tpiyrova 1to1> txo1>v ma Ki>p1a m01x£ia 'T01><; a'U'Ta Ta E1>91)ypaf1f1« 'Tfl'll fl«'Ta 1] 'T� yroviEc; Kat a1t00£1M01>f1£ O'Tl 'Ta 'Tpiyrova £iva1 iaa. A(j)ol) 'Ta 'Tpiyrov � ElV«l iGa Ex01>V oJ..Ec; 'T� 1tUuptc; K«l oJ..Ec; 'T� yrovi£c; 'TO'U<; lG£<; fll« 1tpoc; fll« apa Kal Ta m01xl:ia 1to1> fl«c; .:vota(j)tpo'Uv. EYKAEIAm A' 89 'T.l/27

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A� OOUfJ£ 'tropa 1tOta eivat 'ta 10

Dent ailo Tp(y(I)VU dvut iou

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Kptn]pux ur6n]T� Tpt:yrovrov:

KptTl) pto lGOTTJT<l� ( n - r-n )

Av �vo rpiymva exovv �vo nkvpe, iae, pia npo, pia Kaz 'fr!V nepzexapev17 ymvia rov, iU'fJ,

rare eivaz iaa

e

�� GG r e·

A

2o KptTTj ptO lGOTTt't<l� ( r- n r ) -

Av �vo rpiymva exovv pia 1!Aevpa i{1'f/ Kaz rz, npomceipeve, a'r'f/V nkvpa avrfJ ymvie, iae,

pia npo, pia, rare eivaz iaa.

A'

8

r

R'

r•

A

3 o KptTTj ptO lGOTTtT<X� ( n - n -n )

r

A'

Av �vo rpiymva eyovv rz' nkvpe' rov' iae, pia npo, pia, rare eivaz iaa.

r

r'

Av 'ta 'tpiyrova 1tou 9sA.oufJ£ va <:n>yKpivoufJ£ eivat op9oyroVta 't&tt 'ta Kpt'tl)pta tcr6'trJ't� a1tA.oucr'teUOV'tat aK6J.1a 1ttptcrcr6'ttpo t1tttol) yvrop�ouf.J£ 6'tt sxouv ouo icrt<; yrovi£<;· n<; op9s<;.

'E'tcrt yta va cruyKpivoufJ£ ouo op9oyrovta Tpiyrova XPTtcrtJ.l01totol>f.1£ 'ta 1tapaK<hro KptTI]pta:

� Av �vo opOoycbvza rpiymva exovv �vo avrimozxe, nkvpe, rov, iae, pia npo, pia, rare

eivaz iaa .

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ymvia iU'fJ, rare eivaz iaa.

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llAP A THPH:EEI:E 1) I:t: iaa Tpiyrova antvaV't\ ano ia� nA£upt� IJpi«rKOV'tU\ i«rt:� yrovi£� K«l antvaV't\ ano ia� yrovit:� iat:� nl..t:v pt�. ·

EYKAEIAHl: A' 89 T.l/28

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2) Otav croyKpivouf..t£ ouo -rpiyrova XP110'tJ..lo7toto'6f..1£ Ka9s <popa -ro tva J..lOVo axo -ra KptriJpta tO'O'tll't�, ava:A.oya f..1£ -ra O''tOtXEia xou EXot>f..t£ . Kaxota axo ama -ra O''totXEia 9a oivov-rat axo 'tllV EK<p00V110'11 'tll c; amcrtO'll c; Kat yta 'ta 1)7t0Ao17ta 9a XP110'1J..l07t0t0Uf..l£ yvrocr-rtc; 7tpO'taO'Etc; 'tll c; rsrof..1£'tpiac;, x.x. av EXOUf..l£ tva tO'OO'KEAEc; 'tpiyrovo 'tO'tE �EpOUf..l£ on EXEt ouo iasc; xA.euptc; Kat ouo iasc; yroviEc;.

3) l:E Ka9E KptriJptO tO'O'tll'tac; X,PEta�OJ..laO''tE 'tOUMlXtO''tOV J..tta 7tAzupa. �11Aa0TJ OUO 'tpiyrova xou txouv ne; yroviEc; -roue; J..tia xpoc; J..tia iasc; osv sivat axapai'tll 'ta iaa. fta xapaostyJ..la OUO t0'07tAzupa 'tpiyrova EXOUV 7taV'ta 'ttc; yroviEc; -roue; iasc;, a<pOU Ka9E J..lla a1t0 au-rec; EtVat 60°, alli 01 xA.euptc; -roue; osv sivat xana iasc;. Av iaxus au-ro -ro-rs 9a u7tfJPXE tva J..lOVo taoxA.eupo -rpiyrovo ! ! !

4) llpoaox'l} a-ro 1o KptriJpto OEV 9tAouf..1£ ouo xA.euptc; Kat J..tia yrovia a1J..iJ. axapatriJ-rroc; 'tllV 7tEP1EXOf..I£Vll yrovia.

Ac; A.uaouf..t£ J..ta�i -rropa KaxotEc; aO'Kf)astc; tO'O'tll'tac; -rptyrovrov yta va oouf..t£ xroc; oouA.EUouf..t£ as au-rtc;: 1'1 fl(J'Kif(J1J

l:s taoO"KsAtc; -rpiyrovo ABr f..1£ AB=Ar, <ptpvouf..t£ OtXO'tOJ..lOUc; B�, rE. Na OE�E'tE on Ta -rpiyrova B�r Kat BrE sivat iaa (9a I. J..l7tOpOUO'E va �ll'taEt va OE�OUf..l£ rE=B� f)

'ttc;

A

n.

AUG11 l:uyKpivouf..t£ -ra -rpiyrova B�r Kat BrE. 'Exouv:

1.

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n.

A

Bg = BEr ) Av K -ro 0'11 J.i.t£ o -roJ..tfJ c; -rrov B�, rE -ro-rs -ro -rpiyrovo KEL:l sivat tO'OO"KEAic;.

A

A

B=r

I

erreto� ABr taoaKeAf�

ExoJ.!Evroc; axo -ro 2° KptriJpto r-II-r 'ta 'tpiyrova Eivat iaa.

Br = Br, KOLV� rr'AEvpa A

A

Bl = rl

,

we; �.ll(J'(l (awv YWVLWV aq>ou 8.1, rE

Ms-ra axo Ka9s O'UyKptO'll -rptyrovrov sivat O'll J.a .t vnKo va ypa<pouf..t£ -ra iaa mmxsia (xA.euptc;, yroviEc;) 7tOU 7tpOKU7t'tOUV axo 'tllV tO'O'tll 'ta auriJ, Ei'tE ytari J..l7tOpEi va EiVat O''ta �ll'tOUJ..tEVa 'tll c; amcrtO'll c; Ei'tE ytari 9a J..tac; x.pEtaO''tOUV O''ta E7tOJ..lEVa EproriJJ..la'ta.

'E-rm txouf..t£ : B�=rE, BE=�r Kat ��=Et fta va sivat -ro KEil taoO"KsAic; apKEi va txst ouo iasc; xA.euptc;. E>a os�OUf..l£ on KE=KLl. l:uyKpiVOUf..l£ 'ta -rpiyrova BEK Kat ru. 'Exouv: A

A

El = � I

I

ExoJ.!Evroc; axo -ro 2° KptriJpto r-II-r 'ta -rpiyrova Eivat iaa.

, arro' TO 1 o EpWTTJJl«

BE = ral arro TO 1° EPWTTJJ.l« A

A

B =r 2 2

, , aq>ov, B'LJ1 A f'l:' , w� ptaa, tawv ywvtwv c.

DLXOTOJlOL TWV {awv YWVLWV Bl r avTiaTOLX«

EYKAEIAIIE A' 89 T.l/29

Apa txouv Kat -ra uxoA.otxa a-rotxsia -roue; iaa 07tO'tE KE=KLl.

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DO'f£ o\)o Tpiyrova dva1 ioa

------

"

.L\ive'tat o�eia yrovia xO If/ . L� xA£upt� Ox, O'lf xaipvou� 'ta crrt �ia A,r Kat B, L\ aV'tt<i'totxa, ID<i'tE OA=OB Kat or=O.L\. Na a1tOOE�E'tE on: I. A.L\=Br 11.

Av M 'to crrt �io 'tOfltl� 'trov A.L\ Kat Br 'tO'tE MA=MB "

H OM eivat OtXO'tOflO� Tll � X() If/ H OM eivat �(j0KU9E't0� 'tOU r.L\

Ill. IV.

A1> <J1)

1.

LU-yKpivou� 'ta 'tpiyrova OAL\ Kat OBr. 'Exouv: "

"

0 = 01

KOLV� ywv{a

OA = OB, arro ra &oo11tva r17c; cll1K1'/l11'/c; (� orrwc; AE/lE arro V7l'08El11'/)

or = 0.1, arro ra OE00/1Ev« T1'/c; cll1K1'/l11'/c;

ii. LU-yKpivou� 'ta 'tpiyrova MAr Kat MBA 'Exouv: "

"

MI = M "

2 ' (J)� Ka'taKopu<pt1v "

.L\ B M=r A M , ytati: ax6 1 o epill'tl') fla "

"

Bro = A .l\ 0 Kat MI+.L\BM+M.L\B=1 80° ' M2+rAM+MrA= 1 80° "

f l

Exo fl&Vro� ax6 'tO 1 ° Kpt'ttl pto rr-r-rr 'ta 'tpiyrova eivat iaa, apa "

"

A.L\=Br (Kat OBr = OM ,

ExofJ.EvOO� ax6 'tO 1 ° Kpt'ttl pto rr-r-rr 'ta 'tpiyrova eivat iaa, apa MA=MB (Kat Mr=MB ) iii. ct>tpvou� 'tO eu%-ypafl­ flO 'tfltlfla OM. rta va Oei�ou� "

6n OM OtXO'tOflO� 'tl')�

"

X() If/

apKei va oei�ou� 6n 0 , = 0 2 .LU-yKpivou� 'ta 'tpiyrova OAM Kat OBM. 'Exouv:

ExofJ.Evro� ax6 II-II-II 'ta 'tpiyrova ElVat icra apa Kat

OA=OB,_arro vrr68El11'/

"

OM=OM, KOLv� rrA.Evpa

"

o, = 0 2 iv. . @a

MA=MB, an62° epdJ'f11pa

f.11tOpoucra�

cruyKpivou� 'ta 'tpiyrova orK Kat OL\K Kat va "

OE�OU� 6n rK=KL\ K' M .

=

"

M = 90° 2

Ilapa'tl')po1)� Ofl(J)� 6n 'tO 'tpiyrovo orL\ eivat tcrOcrKEAi� a<pm) Or=O.L\ Kat ax6 'tO 1tPOll'YO'UfJ.EVO epiD'tl')fla OM OtXO'tOflO� xou avncr'totXEi cr'tl') �licrrt apa eivat Kat 1>vo� Kat Ota�cro� 'tOU 'tptyrovou E1tOflEv(J)� Eivat KU9E'tl') <i'tl') rL\ Kat ottpxe'tat ax6 'tO f.l£cro 'tl')� ox6'te eivat �<iOKU9E'tO� 'tOU r.L\.

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3" aUK1fl11f l:E cido J..lE

KEvtpo 0 , 9EropOUJ..LE 't11 xopoi] AB Kat ouo <JT)J.I.Eta r, ll navro <J't1')V AB 'tE'tOta romE Af=B!l. Na anoOE�E'tE 6n: I. Of=O!l 11.

Av M -ro J.ttcro -rou rD. OA=OB,_w� a�crlvE� rov lowv Kvclov Kat K -ro <JT)J.tEio crm onoio T) OM 'tEJ.lVEt -rov ciKA.o -r6-rE -ro Af=B-1, cmo VTC08EUrJ -rpiyrovo AKB Eivat tcrocrKEAi�.

1\

Au<f11

1\

I. «PtpvouJ..LE n� aKtivE� OA, OB Kat cruyKpivouJ..LE -ra -rpiyrova OAf, OB!l. 'Exouv:

A = B , ETrEL6tj ro AOB rpivwvo Eivat LCJOCJKEMc;

Kt

Eno�ro� 'ta -rpiyrova an6 n-r-n Eivat icra cipa txouv icra Kat 'ta '1)1t0Aot1ta (J'totXEia 'tO'I)� t-rm Of=O!l.

}

An6 'tO 1tp0T)'YOUJ..LEVO EpcO't1')J.ta tcrx\)Et Of=O!l cipa 'tO -rpiyrovo o r!l EtVat tcrOcrKEAe� Kat a<pou M J.tEcro 'tll � f!l T) OM AM= BM,_a<pov AM=Ar+TM=Bil +LlM=BM. Eivat otciJ..LEcro� cipa Kat U'JfO�.

11.

{

·

OK=OK, Ko tv� nA.Evpa

1\

1\

'E-rm txouJ..LE Mt = M2 = 90° . fta va oEi;oUJ.tE 6n 'tO AKB tcrocrKEAe� apKEi va OE�OUJ..LE 6n EXEt ouo icrE� nA.rnpt�. l:uyKpivouJ..LE -ra op9oyrovta _-rpiyrova AKM Kat BKM. 'Exouv : Eno�ro� -ra -rpiyrova Eivat icra, cipa AK=BK.

AuKijuet� yta E'auK1Jt1'1J

1 . !livE-rat tcrocrKEAe� -rpiyrovo AB f J..LE AB=Af. An6 -ro <JT)J..LEtO B <ptpvotiJ.tE Kci9E't1') cr't1')V AB Kat an6 'tO r Kci9E't1') cr't1')V Af. Ticivro crn� Kci9E'tE� 9EropOUJ..LE 'ta <JT)J..LEta !l Kat E av-ricr-rotxa rocr-rE B!l=fE. Na anoOEi;uE 6n: i.

A!l=AE ii.

Av K, A -ra <JT)J..LEta ma onoia T) !lE 'tEJ.lVEt n� AB, Af av-ricr-rmxa va OE�E'tE 6n -ra -rpiyrova A!lK, AEA Eivat icra.

111.

Ta <JT)J..LEia K, A tcrantxouv an6 't11 Bf. 2. !livE-rat -rpiyrovo ABf -rt-roto rocr-rE AB<Af. Ticivro cr't1')V Af naipvouJ.tE <JT)J.tEio !l -rt-roto rocr-rE All=AB Kat <J't1')V npoeK'ta<JT) 'tll � AB npo� -ro B <JT)J..LEtO E t-rm rocr-rE AE=Af. Na OE�E'tE 6n: 1.

E!l=Bf 11.

Av K -ro <JT)J..LEio 'tOJ.lll� 'tll � Ell J..LE 't1')V Bf -r6-rE -ra -rpiyrova BKE Kat !lKf Eivat icra 111.

Ta- -rpiyrova r!lE Kat BEf EtVat icra 3. An6 -ro �cro M 'tll � uno-rEivoucra� EV6� op9oyroviou -rptyrovou -ra Kci9E-ra EU9UypaJ.tJ.ta 'tJ.lllJ.la'ta M!l Kat ME 1tp0� 'tll � nA.rnpt� AB Kat Af avricr-rotxa. Na oEi;E'tE 6n: 1.

Ta -rpiyrova BllM Kat MEf Eivat icra ii. !lE otxo-r6J.to� 'tll � yrovia� A!lM 4. !livE-rat -rpiyrovo ABf J..LE AB=2Af. Av All 11 OtXO'tOJ.lO� 'tll � yrovia� A Kat M -ro J.tEcro 'tll � AB, va OE�E'tE 6n: 1.

Af=AM 11. llf=!lE All Kci9E't1') <J't1')V fE 111.

EYKAEIAHI: A' 89 T.l/31

Ma8r}tJOTIKOi 111ay6lVIOIJOi Em.ptM:t.a: Ent.'Tpom] At.ayrovt.aprov

1 7 11

BaAKavLK� M a8 rnJ.a"tLK� OAu �-tnuxoa N f:wv Tou pK(a, AnitAELa,

2 1-26 l ouv(ou 2013

H 1 711 Ba.A.KavucfJ Ma9TJJ.Wnldt OA.uJ.tmaoa Nerov o�ftx9TJ CJTllV AttciA.eta Til e; ToupKiac; a1t6 21 J.ttXPt Kat 26 Iouviou 20 1 1 J.t£ Tl1 <JUJ.lJ.t£'toxrl l 1 3 J.ta9TJtrov Kat J.ta9TJtptrov. 'EA.apav J.ttpoc; EmcrytJ.troc; 1 1 xropec;: AA.Pavia, Bocrvia Kat Ep�eyoPiVTJ, BouA.yapia, EUcloa, K61tpoc;, Maupopouv10, MoA.Oapia, IIrdM, POUJ.tavia, I:eppia, ToupKia Kat avemcrftJ.troc; Ot xropec;: A�EpJ.l1tO.'it�av, fa.A.A.ia, HvroJ.ttvec; lloA.ttEiEc; AJ.ltptJdtc;, lvOoVTJcria, Ka�aKmav, I:aouotKft Apapia, Tat�tKtcrtav, ct>tA.m1tivec; Ka9roc; Kat TJ B ' OJ.laOa Tile; ToupKiac;. TTJV EUTJvtKit OJ.laOa a1toteA.oucrav ot J.la9TJtec;: Mda� Al)pl]Tpt.o�, Apyupo JU:Tallt.o Apyupo Jl.f:Tallw MunaKo� llavay«<'TTJ � XaA.Kt.vo ptTallw BEV�tA.o� rtropyt.o� XaA.Kt.vo Jl.f:Tallw KaA.oaw11� Nt.KoA.ao�, I:1>TlAW1)� ClliA.t.1t1tO� - IA.apirov, Ei)(p1)p1) pvda I:1>pp£TOXll K£Ta£Taw1)� Paq»al]A., APXI1r6c; Tile; EUTJvtKitc; OJ.laOac; i}tav o J.ta9fJJ.tanK6c; llfTpo� M1tpfrt.aVVTJ� Kat u1taPXIlr6c; o J.ta9TJJ.tattK6c; I:t.A.o1>avo� M1tpa�t.TiKo�. Ot J.ta9fJt£c; otayrov�ovtat cre tecrcrepa 1tpopA.ftJ.lata 1tou Ka9eva a1t6 auta .paeJ.toA.oyeitat J.t£ 1 0 J.lOVooec;. I:uJ.tq>rova J.t£ TllV aro1tTJ paeJ.toA.oyia 11 EUcloa J.t£ 1 22 J.tovaoec; KateA.ape TllV 511 9E<JTJ J.t£ta9) 't(I}V E1ttcrytJ.troc; <JUJ.lJ.lt'tEXOU<JIDV xroprov, EVID i}tav CJ't11V 811 et<JTJ J.lt't�U 't(I}V 20 OJ.lclO(I}V 1tOU CJUVOAtKcl tA.apav J.ttpoc;. Ttc; 'tECJ<JEptc; 1tprotec; etcretc; Kat&Pav Ot OJ.lclOE<; Tile; PoUJ.taviac; ( 1 99) , ToupKia ( 1 97), I:eppiac; ( 1 68). BouA.yapiac; ( 1 65).

Tu npopl.. iH JUTfl.

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n po filclJ !-lU I . Bptt'TE ola 'TU OIU'TE'Tayptva �E\Yy1) a, b 9E'TIKIDV UKEpairov 1t01> dvat. d'TOJ.a

romE 01 apt.9poi At>()lJ

3 a b-1

-­

a+l

KUI

3 b a+l b-1

, � , ;. , VU EIVUI KUI 01 ut>O 9 E'TIKOI UK«:.pUIOL

{ ) ) ( ) I{ a3 + 1 ) , yta va eivat o a�:� 1 9enK6<; aKepato<; 1tpE1tEt Kat apKEi ( a + 1 )l ( b + 1 ) . E1tetoft b 3 a - 1 = a { b 3 - 1 ) + ( a + 1 ) Kat ( b - 1 )i { b3 - 1 ) , yta va b3 a + 1 , , , , 9 E'tlKO<; aKEpatO<; 1tpE1tEt Kat apKEt' ( b - 1 )1 ( a + 1 ) .

Etvat 0 b-1

E1tet0it a 3 b - 1 = b a 3 + 1 - ( b + 1 Kat a + 1

E1to�ro<; yta Eivat Kat ta oi>o dacrJlata 9EnKoi aKEpatot 1tpE1tEt Kat apKEi va tcrxUEt: ( b - 1 I( b + 1 , 01t6-re E1tEtat 6n 1tp£1tet: ( b - 1 I( b + 1 + b - 1 => ( b - 1 12 , ortA.aoft 1tpE1tEt

)

)

b - 1 e { 1, 2} ::::> b e { 2, 3 } .

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•

)

)

)

( a + 1 ) lb + 1 = 3 => a = 2 , o1t6-re ( a, b ) = ( 2, 2) Eivat J.lOVaotKft A.i>crrt crtrtv 1tEpi1ttro crrt autft. fta b = 3 exm>JlE 6n: ( a + 1 )l b + 1 = 4 => a = 1 it a = 3 , 01tO'tE ( a, b ) = ( 1, 3 ) Kat ( a, b ) = ( 3, 3 ) eivat ot J.lOVaOtKE<; A.i>cret<; <JtrtV 1tEpi1ttro<Jrt autft.

11

fta b = 2 txouJ.tE 6n:

EYKAEIAm A' 8 9 T.l/32

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llpofl A.l) Jl « 2 . 'E<rrro ABC o;uymvao Tpiymvo fl£ AB < AC Kat 0 To KtvTpo TO'U 1tEptyqp«J.tJ.ttvO'U K6KA.o1> TO'U ro 'EGTro D G1Jfldo navro <rro E1>06yp«JlJlO TJ.t'I\J.t« BC TtToto •

cO<rrE BAn = CAO. 'E<rrro E TO o£6't'Epo aqpdo TOJ.t'l\� TO'U K6KA.o1> r.o Kat 't"'� E'U8Eia� AD M, N Kat P dvat T« J.ttaa Trov E1>8uypapprov Tpl)paTrov BE, OD Kat AC, «VTi<rrmza, va anooEi;ETE OTI. T« G'IJ!El« M, N Kat P dvat G'UVE1>8Et«Ka. Ai>Gl) (1 °� Tp01t0�) ea <l1tOOEi�O'UJ.lE 6n TO TE't'pcl1tAEUpo MOPD sivat 1t<lp<lMT)AOyp<lJ!JlO, <l1t0 TO orroio 1tp01cU1tTEt on T<l crll J.tEia M, N Kat P sivat crovw9staKa. Errstoi)

•

Av

.

-

Eivat BAD = CAO = 90° ABC, TO D Eivat TO ixvo� TT)� Kcl9ETT)� arr6 TO A 1tp0� TTJV 1tAEupa BC . E1tstoi) TO M sivat To J!Ecro Tou su9l)ypaJ.tJ.tOU TJ.1TJJ.1<lTO� BE , sxouJ.tE 6n

BM = ME = lvfD , o1t6TE A1iJE = A1£D = ACB . 'Ecrro 6n TJ wesia lvfD TSJ.!VEt TTJV wesia A c crTO Dl .

Afi>E = lv.IED = A CB, TJ wesia lvfD sivat Ka9ETTJ 1tpo� TTJV AC. EmrrMov, E1tEtoi)

E1tstoi) :

ADDI

=

To 0 sivat To 1tspiKEVTpo Tou Tptyrovou ABC Kat TO p sivat TO J.!Scro TT)� 1tAEupa� A c ' TJ wesia

OP Eivat Kcl9ETT) 1tp0� TTJV wesia AC . Apa 01 wesis� lvfD Kat OP sivat 1t<lpclMTJAE�.OJ.1oiro�, E1tstoi) To P sivat To J!Ecro TTJ� 1tM:upa� A C, SXOUJ.lE AP = PC = DP ' 01tOTE PDC = A CB .

'EcrTro Tffipa TJ wesia PD TSJ.!VEt TTJV suesia BE crTo CJl1J.1Eio D2 Errstoi) •

A

A

A

A

BDD2 = PDC = A CB = BED srrsTat 6n TJ w9sia PD sivat Ka9ETTJ rrpo� TTJV w9Eia BE . Errstoi) TO M sivat TO J!Ecro TOU weUypaJ.tJ.!OU TJ.1TJJ.1<lTO� BE , TJ wesia OM sivat Kci9ETT) 1tp0� TTJV wesia BE ' 01tOTE 01 wesis� 0M Kat pD sivat 1t<lpaMTJAE�.

2°� Tp01t0�

E1tstoi) Eivat BAD = CAO = 90° - ABC, TO D Eivat TO ixvo� TTJ� Ka9ETTJ� a1t6 To A 1tpo� TTJV 1tM:upa BC . 'Ecnro H to op96Kevrpo tou A B C. Tote yvropi�ouJ.t& 6n Hl) • l)B, EnoJJ£vro� crto tpiyrovo 1111 11 MC crovoeet ta J.ttcra, enoJJ£vro� ,3r,' l) 11 BH Kat MD • ¥- • OP. A.pa- txouJ.t& M O ll • 0/!, EnoJJ£vro� to tetpanJ.eupo MDI'O eivat napaUT)MypaJ.lJ.lO ..

Kat TO �TJTO'I)J.t&Vo en&Tat.

llpofl A.ll J.t « 3 . Na anooEi;ETE O't't

(

2a + 2b + -a+1

)(

)

2- � 1 6, b + 2 a + -b+1

8ETtKOU� «KtpatO'U� a K«t b TtTOtO'U� cOGTE ab � 1 .

Ai>Gl) (1 °� Tp01t0�) A1t6 TTJV avtcr6TTJT<l apt9J.tTJTtKoi> - ysroJ.tETptKoi> J!Ecrou, a<pm)

yt« oA.o1>� TO'U�

a, b esnKoi aKspatot, sxouJ.tE: a + 3 + 2b a+l+ 2 ;::: 2�a+2b+ . 2 =-+-+ a+l 2 a-1 +2b ;::: 2+ a-I +2b a + 2b + 2 ;::: -2 a+l a+l 2 a+l 2 2 a+l 2

-

-

-

-

-

EYKAEIAHl:. A' 89 -r.l/33

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--

(1)

------ Ma9f1p«TtKoi Atay(l)vtopoi

2 ;;:: -b + 3 + 2a b + 2a + -b+1 2

------

(2)

( 1) Kat (2) f.tS noU.anA.acrtacrf.t6 Ka'tci f.tEATt A<lf.t�civouf.!S a+3 +'lhl(b+3 +2al =>(a+ 2h +2)(b+2a +2) > ( a+4b +3)( b+4a +3) (3 ) 2a,._.2_);;:: (a+2h? )(b+ ( a+1 b+1 4 a+1 b+ 1 2 ) 2 ') EmnMov, ano 'tllV avtcrO'tll't<l Cauchy - Schwarz Kat OtOOf.tSvOU on ab ;;:: 1 , sxouf.tt (a + 4b + 3) ( b + 4a + 3) ;;:: ( .J;;b + 4.J;;b + 3 ) ;;:: 64 , (4) on6'tt ano ne; (3) Kat (4) E1tt'tat l1 �ll'tOUf.tSVTt avtcrO'tll't <l.

1 (2°-; Tpono�) : Enstoi) a,b 9snKoi Kat ab ;;:: 1 sxouf.tt a + b ;;:: a + .!. ;;:: 2.Ja · = 2 , on6'tt a a

An6 ne;

2

)2

2 = b + (a + b) +2 ;;:: b + 2 +-;;:: 2 b+1 +b+1 +1+2 ;;:: 4 4 ( b + 1 a + 2b + a+1 a+1 a+1 2 2 a + 1 2(a + 1) 2 ;;:: 4 4 (a + 1l b + 2a +b + 1 2(b + 1)

Of.toiroc; A<lf.t�clVOUf.!S

(5)

(6)

(6) f.tS noM<lnA.amacrf.t6 KaTci f.tEATt A<lf.t�clVOUf.!S 2- ) ( b + 2a + -2-) ;;:: 16 4 (a + 1)(b + 1) ;;:: 16 4 ( 2Fa ) ( 2 .Jb) ( a + 2b + b+1 4 a+1 4 => (a + 2b + -2-)(b + 2a + -2-) = 16v;;h ;;:: 16. a+1 b+1

An6 ne; ( 5) Kat

(3°.; Tp ono�) : An6 'tllV avtcrO'tll T<l Cauchy - Schwarz EXOl>f.tS

(

(a+2b+�) a+1 (b+2a+2)=((a+b)+b+�)((a+b)+a+2) b+1 a+1 b+1 ;;:: a+b+ J;;b + { a+1ub+1 EmnMov, ano 'tllV avtcr6'tll 't<l apt9f.tTtnKou - ytrof.!STptKou JJ.Ecrou A<lf.t�civouf.!S

)J2 •

2 >- 4 (� �(a + 1)(b + 1) a + b + 2 4 = (a+b+1)(a+b-2) +4 ;;:: 4, 2 ono'tt sxouf.!S a+b+ vab + ;;:: a+b+ 1 + I a+b+2 ( a+b+2 ) v(a+1)(b+1) a<pou a + b ;;:: 2.rah ;;:: 2 . An6 ne; crxscrstc; (7) Kat (9) S1tt't<lt l1 �ll'tOUf.tSVTt avtcrO'tll 't<l.

(7)

'

,

,

r;-

(9)

llpopA.ru.ta 4. 'Ecnro n tva� 9ETtK6<; aKtpato<;. A'6o naiKTE<;, 11 .AJ.iK1] Kat o Ba<ril11<;, na�o1>v TO aKOA01>9o natyviot: • H AliK1] E1tlA.tyEt n npaypanKo'6<; apt9po'6�, 6xt KaT' av6.yK1] ota<popETtKo'6<;.

• H AliK1] GT1] Gl>VExEta ypa<pEt TU a9poi<rpaTa olrov TO>V �El>yrov TO>V 1tapanavro apt9prov

navro GE pia Kolla xapTi T1]V onoia oivEt cno Ba<riA11. (Ynapxoov •

(

n n-1

2

)

TETOta

a9poi<rpaTa, oxt KaT' ava"(K11 Ota<popETtKa) 0 Baail11<; KEpoi�Et, av pp�:t <rrocna TOl><; n apt9po'6<; no1> «PXtK6. Eix�: E1ttlt�Et 11 AliK1].

Mnopd o Baail11<; va dvat ptpato<; OTt 9a KEpoi<rEt cnt<; aKOAo1>9E<; 1tEpt1tTID<rEt<;; c. n = 8 b. n = 6 a. n = 5 AtKatoloyEicnE Tt<; anavn]<rEt<; <ra<;.

[Fla napab&l"/Jla, ot"av n = 4, 'I AUK1! pnopei va e1t1U�e1 'l'OV� ap18pov� 1, 5, 7, 9 nov ixovv EYKAEIAHI: A' 89 T.l/34

------ MaOqpaTtKoi Ataymvuspoi

------

-ra i�1a afJpoiu}la-ra JIB -rove; aplfJ}Iovc; 2,4, 6,1 0 Kal trm o Bauil'lc; �ev JI1COpei va eiva1 ptpawc; o-r1 fJa Kep�iuel.)

At> <Jll (a) H a1tUVtTJOTJ sivat vat. 'Ec:rtro a � b c d apt6f.loi 1tou E1tsM:i;s 11 AAbcrJ. E1tetOij Ka6s apt6f.l6� ef.l<pavii;E'tat crs a6pOtcrf.!a i;&Uyou� 'tscrcrspt� <pops�, f.lE 1tp6cr6e<J11 'tffiV 1 0 Ota<popsnKrov t;suyrov Kat OtaipEOTJ 'tOU a1tO'tEAScrf.!a'to� f.lE 'tO o BacriA11� Mxf.!Pavst a b c d . A<patprov'ta� a1t6 am6 'tO sA.axtcr'tO Kat 'tO f.lS'Yl(j'fO a6pOtcrf.!a i;euyrov a b Kat d Aaf.!PUVE1 'tO c . A<patprovm� 'tO c a1t6 'tO OW'tepo f.lE'YaAU'tepo a6pOtcrf.!a t;euyrov c+ Aaf.!PUVE1 'tO . A<patprov'ta� 'tO (l1t0 'tO f.lE'YaAU'tepo a6pOtcrf.!a seurrov d A.af.!Pavst 'tO d . Of.loiro� f.l1tOpei 6 BacriA11� va 1tpocrotopicrst 'tOU� apt6f.lOU� a Kat b . (b) H a1tUVtTJOTJ sivat vat. 'Ecr'tro a � b c d f apt6f.loi 1tou E1tsM:I;s 11 AAiKll . E1tet0ij KU6e apt6f.l6� Ef.l<pavii;e'tat cre a6pOtcrf.!a t;&Uyou� 1tME <pops�, f.lE 1tp6cr6e<J11 'f(I)V Ota<popsnKrov t;euyrov Kat OtaipEOTJ 'tou a1tO'tEAScrf.!a'to� f.lE 'to o BacriA11� A.af.!Pavst a+b c d f . A<patprov'ta� a1t6 am6 'tO EAUXtcr'tO Kat 'tO f.!Sytm;o a6pOtcrf.!a i;euyrov a b Kat f l..af.!Pavst 'to c d . A<patprov'ta� 'to sA.axtcr'to Kat 'tO OW'tepo f.!E'YnAU'tepo a6pOtcrf.!a i;euyrov a b Kat d j a1t6 'tO a b c d j Mxf.lPUVEt 'tO c Kat Of.!Otro� f.l1tOpei va A.apst 'tO b d . I:tTJ cruvsxsta 6a XPllcrtf.l01tatijcrst 'ta 1tapa1tavro i;s011 yta va A.UPst 'ta sWrll a + f Kat b Tropa 'ta a6poicrf.!a'ta b, c, b c sivat 'ta 'tpia f.l1Kp6'tepa f.!E'tal;u 'trov a6potcrf.!a'trov i;euyrov. Av o BacriA11� 1tpocr6scrst au'ta 'ta 'tpia, a<patpscrst a1t6 'to a1to'tsAecrf.!a 'ta yvrocr'ta a6poicrf.la'ta c + d Kat b Kat Otatpscrst tTJ Ota<popa f.lE 'to 'tO'tE 6a t..apst 'tO a . I:tTJ cruvsxsta a1t6 'ta yvrocr'ta a6poicrf.!a'ta Mxf.!Pavet 6A.ou� 'tOt>� t>1t6A.omou� apt6f.lOU�. (c) H a1tUVtTJOTJ Eivat oxt. Av .., AAiKll e1ttM/;et 'tO'O� OK'tOO apt6f.lOU� 9, 20, 'tO'tE o BacriA11� oev f.l1tOpei va sivat criyoupo� 6n 6a 'tOU� ppst Ka6ro� Kat apt6f.loi 6, 9 oivouv aKptpro� 'ta iota a6poicrf.la'ta.

� � � em

4,

+ + + +e +e e

e

+ + +e+ + e+ 15

e � � �e� m

+ +

+

+

+ + + +e+

+e.

+e

a+ a+ + +e

2

1, 5, 7, 12, 14, 16, m 2, 4,

28

IIPOBAHMA 1 x, y

15

5,

10, 1 1, 15, 17, 1

Av ot

+ +e

npoKpltJ.attK6c; 6tayWVLOtJ.O<; NE:wv 27 AnptA(ou 2013

£iva1 0£'ttKoi xpay11anKol ap.0110i, va wtoi•'9m On:

rt.a 1tOt£.; TlJ.IE.; 'T(I)V x, y l<JXV£l ll l<J01'11T «; AV <Jfl (1oc; Tpono.;) E1tst0ij sivat > 0 11 osoof.lSvll avtcrOtlJ'ta sivat tcroouvaf.!TJ f.lE tTJV

( :)(� 2);, x+

+

8.

x, y (xy + 2)(y + 2x) ;;::: 8xy � xy2 + 2y + 2x2y + 4x - 8xy ;;::: 0 � {xy2 -4xy + 4x) + (2x2y - 4xy + 2y) ;;::: 0� x(y2 -4y + 4) + 2y(x2 -2x + 1) ;;::: 0 � x(y - 2) 2 + 2y(x - 1)2 ;::: 0, 1tou tcrx\)st a<povx,y > O. H tcr6tlJ'ta tcrx'l)st 6'tav : x(y - 2) = 0 Kat 2y(x -1) = 0 � x = 1, y = 2, a<pot> x,y 0. >

(2°c; Tp07Co.;) fpa<pOUf.lE tTJ OEOOf.lSYTJ avtcrOtlJ'ta crtlJV tcroouvaf.!TJ f.lOp<pij tTJ�

( xy + 2 )( y + 2x) ;;::: 8xy

(1)

0sropOUf.lE tTJV yvrocr'tij avtcrOtlJ'ta apt6f.lTJ't1KOU - 'YEOOf.!E'tptKOU f.!Scrou yta a, b ;;::: 0, (TJ tcrOtlJ'ta tcrx6et 6'tav a = b ) tTJV 01toia E<papf.!OSOl>f.lE Ouo <pops� , yta a = = Kat yta a = = 01tO'tE Mxf.lPUVOUf.lE

a + b ;;::: 2J;;b,

xy,b 2

EYKAEIAHI: A' 89 T.l/35

y,b 2x ,

------- Ma91)JUl'TlKO( AtayrovtCJftO(

-------

2 � 2jhY ' y 2x � 2jhY ,

(2) 61tou 11 t<JOtT)'ta tcrXt}et 6'tav xy = 2 , Kat (3) 61tou 11 tcr6tTJ'ta tcrXt}et 6Tav y = 2x . E1tetSi] Kat 'ta Suo �A-11 'tOlV avtcro'ti]Trov (2) Kat (3) eivat xy + +

9enKa, xy +

(

JlE 1tOMa1tAacrtacrf.L6

2) ( y + 2x)

amrov Ka'ta flEA11 A.aflP<lvouJ.tE � Sxy . H tcr6tT)'ta tcrXt}et 6'tav xy Kat

=2

ll PO B A H M A

y = 2x , S11A,aSi] 6Tav x = 1, y = 2.

2 0t:copo'6pt: 1tavco <JE K'6KJ..o n ota(J)opt:TtKa all JtEia, TETota romt: va Jll]V 1>1taf1Xovv TPE� XOPO� 1t01> opi�OVTCll Cl1t0 Cl'UTa TU ai]JIE{U 1t01> VU 1tEpvot)v Cl1t0 TO i010 ai]J1El0 <JTO E<JO>TEplKO TO'U K'6KAO'U. . («) Na ppEiTE Tl]V TlJITt TO'U n, «V 0 apt9po� TO>V TptyroVO>V 1t01> opi�OVTCll JIE KOp'U(J)E� Tp{a a1to T« aqpda t<J0'6Tat J1E 2n. (p). Na ppEiTE Tl]V TlJITt TO'U n, QV 0 apt9po� TO>V ai]JlElO>V TOJ1Tt� TO>V zopOIDV 1t01> ppiGKOVTCll <JTO E<JO>TEplKO TO'U K'6Jdo1> l<J0'6T«l J1E 5o.

n

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n

(a) 01totaSi]1tO'tE 'tpia a1t6 'ta <Jflflcta 1tOU ppicrKOV'tat 1tUVOO <J'tO KUKAo opil;ouv €\la 'tpiyrovo. E1tofl€vroc; o apt9f.l6c; 'tOOV 'tptyrovrov 1tou opi�oV'tat a1t6 'ta attJ.tEia eivat icroc; flE 'tOV apt9fl6 'tOOV Stac:popenKrov u1tocruv6A.rov flE 'tpia cr'totXEia 1tou fl1tOpouJ.tE va E1ttAE�OUflE a1t6 'tO crUVOAo 'tOOV n <JflflEtOOV. 0 apt9f.l6c; aU't6c; t<JOU'tat JlE 'tO crUVOAo 'tOOV cruvSua<JfliDV 'tOOV crtotXtirov avU <pia, i5T]Ml0ij clvat o

n

n

(;) = n (n - I�(n - 2) .

n(n-1)(n-2) = 2n ( n -1 )( n-2) = 12 n -1 = 4 n = 5 .

6 Stac:popenKrov crtt J.tEirov a1t6 'ta n attJ.tEia 1tou ppicrKOV'tat 1tavro

E1tofl€vroc; sxouJ.tE tTJV e�icrrocrtt

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·

<:::>

(�) 01totaSi]1to'te 'tE'tpaSa crTov K'6do opi�et KUp't6 'tE'tpa1tAeupo 'tOU o1toiou ot Stayrovtec; 'tEflVOV'tat cre €\la ecrro'teptK6 <Jflf.lEio 'tou K'6dou. AvTicr'tpoc:pa , o1totoSi]1tO'tE ecrro'teptK6 crtt J.tEio 'tOU K'6clou 1tOU Eivat crtt J.tEio 'tOf.lf]c; Suo xopSrov 'tOU opi�Et f.Lta 'tE'tpaSa Stac:popE'ttKIDV <JfiJ.!EtOOV 'tOU KUKAOU 1tOU eivat 'ta aKpa 't(J)V xopSrov amrov. E1tOflEvOOc; 'tO crUVOAo 't(J)V E<JOO'tEptKIDV <:IT}flEtOOV 'tOU K'6dou 1tOU eivat crtt J.tEia 'tOf.lf]c; Suo xopSrov Kat ppicrKOV'tat <J'tO E<JOO'tEptK6 'tOU K'6dou Eivat ·

n( n -1)( n -2)( n -3) = 5n ( n -1)( n - 2) { n - 3) = 120 .

24 ( n -1)( n - 2)(n -3) = 23 3 5 = 4 · 5 · 6 n - 1 = 6 n = 7.

E1tofl€vroc; EXOUJ.!E �

(4n) = 4!·(nn!-4)! = n(n - 1)(n24- 2)(n - 3) <:::>

·

·

<:::>

<:::>

llP O BAHMA 3

Eivat 1tp<OTo� 9t:TtKo� aKtpato� Kat ot x, y Eivat 9t:TtKoi aKtpatot, va ppEiTE, .

(J't)VUp'rft <JEl TO'U p , OAa TU �E'Uyapta ( X, y) 1t01> ElVCll J..'6<JE� Tl]� t:;i<JO>al]�: Av

p

p ( x - 2 ) x ( y - 1) .

(1) AV E1tl1tUOV OiVETCll OTl X + y 21 ' ppEiTE oi..E� T� TptaOE� ( x, y' p) 1t01) Eivat J..t><JE� Tl]� =

=

t:;i<JO>al]� (1).

A6att

x- 2 = 0 x = 2, 'tOTE 1tpE1tEt y = 1 Kat 11 e�icrrocrtt sxet tTJ AU<Jfl (x, y) = ( 2,1) .

Av Eivat x-2 0 , 'tOTE, E1tttSi] o p eivat 1tpro'toc; 9enK6c; aK€patoc;, a1t6 tTJV i] pl y -1. p(x - 2) = x(y -1) 1tpOKU1t'tEt 6'tt: y -::t= 1 Kat p l x Av

�

-::�=

EYKAEIAHI:. A ' 89 -r.l/36

e�icrrocrtt

------ MuOflfUl'ttKoi Atuyrovurp.oi •

•

•

------

p l x, 'tO'tS X = px'' 01tOU x' esnK6� aKtpato� ' 01tO'tS 11 s�icrro<f11 yivs'tat: p(px' -2) = px'(y - 1) � px' - 2 = x'(y-1) � x'(p -y + 1) = 2 � x' = 1, p -y + 1 = 2 i] x' = 2, p -y + 1 = 1 � x' = 1, y = p -1 t1 x' = 2, y = p � x = p,y = p -1 i] x = 2p,y = p � (x,y) = (p,p -1) 1j (x,y) = (2p,p) AV p IY -1, 'tO'tS y -1 = py'' 01tOU y' esnK6� aKtpato�. 01tO'tS 11 s�i<rOO<f11 (1) yivs'tat: p (X- 2) = Xpy' � X- 2 = xy ' � X(1- y') = 2 � X = 1, 1 -y' = 2 tl X = 2, 1 -y' = 1 � �x=1,y' =-1 (o&JvarrV i] x=1,y' =0 (o&Jvatll). Av sntnA.tov tcrx;6st 6n x+y=21, 'tO'tE txoufJ£: AV esropi]crOUJ.lS n� A.ucrst� (X, y) = ( p' p -1) ' 'tO'tS 2 p -1 = 21 p = 1 1 ' 01tO'tS 1tpOKU1t'tSt 11 'tpui�a ( x, y, p) = ( 1 1, 10, 1 1) .

Av esropi]crOUf.l£ � A.ucrst� (.x; y) =( 2p,p) ' 'tO'tS 3p=21c:p=7, 01tO'tS 1tpOKU1t'tSt 11 'tptci�a ( .x;y,p) =( 1'\ 7, � .

AV

�

n POBAHMA 4 AivE'Tat K'6Jdo� c(O, R) (pE KtvTpo TO <J11 JI.ElO

0

Kat aKTiva R), pia ouipETp6� TOt>

AB

Kat 'TO JIEGO r'Tot> 'T6;ot> AB. I:.'Tll G'UVfxEta ypti<pOt>JIE K'6Jdo cl (K, KO) (JIE KEv'TpO <J11 JI.El0

K 'TOt> 'TJI'I\Jia'TO� OA Kat aK'Tiva KO < R ) Kat 9sropo'6pE 'T� E<ptt1t't'6JIEVE� r.A , ro a1t6 'TO <J11 JI.El0 r 1tpo� 'TOV K'6Jdo c1 (K, KO) . H Et>9Eia K.A 'TEJIVEt 'TOV K'6Jdo c(O, R) GTa G1]pEia E

Kat z ('TO <J11 JI.El0 E ppiGKE'Tat G'TO 11J1tK'6.U..to 1tO'U av1]KEt 'TO <J11 JI.El0 r ) . Ot Et>9EiE� 'TEW� Kat rz 'TtpVO'UV 'TllV AB G'Ta G1]pEia N Kat M aV'TiGTotxa. Na tt1tOOEi;E'TE 6'Tt 'TO 'TE'Tpti1ti.E'Upo EMZN sivat tGOGKEAE� Tpa1tt�to, syysy papptvo GE K'6Jdo TOt> o1toiot> TO KMpO ppiGKE'Tat E1ttlV(l) G'TOV K'6Jdo c(O, R) .

Er

A'6a11

r

'Ecr'tro 'tpiyrovo

E1 = rEZ , �

�

D1E

H yrovia

'tO'tE an6 'tO opeoyrovto

t

txoufJ£:

1 = 90° -E1

(1)

•

roz = 01 sivat 11 av'ticr'totXJI �

�

E1 = FEZ , on6'ts 01 = 2E1 Kat FOZ EXOUfJ£: t 2 180° -61 180° ....:. 2 £1 = 2 = 2 = 900 - E:� . (2) A1t6 't� crxtcrst� (1) Kat (2) , EXOUfJ£: � = /; . :EuyKpivoufJ£ 'tropa 'ta 'tpiyrova FL1E Kat :ExflJ.ta 1 FOM , yta 'ta onoia txoufJ£: 1) Eivat opeoyrovta ( r&J = rOM = 9(! ) . 2) 'Exouv J.tia o�sia yrovia i<r11 ( � = f2 ).

3) 'Exouv J.tia Kaes'tll nA.supa i<r11 (FL1=F0, ro� s<pan't6J.tEVa 'tJ.ltlJ.la'ta cr'to K:UK:A.o (c1) , an6 'tO <f11 f.l£t0 r ) . Apa 'ta 'tpiyrova sivat FL1E ' roM sivat icra Kat crs cruv�ua<rJ.lO f.l£ 'tllV t<JO'tll 'ta 'tOOV opeoyrovirov 'tptyrovrov F.KL1 Kat FKO , <rUJ.l1tSpaiVOUf.l£ on 'tO 'tpiyrovo FEM sivat tcrocrKsA.t� Kat 11 IK sivat �tXO'tOJ.lO� 'tll � yrovia� EfM Kat J.lScroKaes'to� 'tll � EM . (A) An6 'tllV t<JO'tll'ta 'tOOV opeoyrovirov 'tptyrovrov FL1E Kat roM ' <rUJ.l1tSpaivOUf.l£ 6n: r.Ez = r.Mo ' 01tO'tS Kat ot napa1tA11POOJ.la'ttKE� 'tOU� yrovis� ea sivat icrs�. �11Aa�i] NEZ = NMZ . Apa 'tO 'tE'tpanA.supo EMZN sivat syypa\jltJ.lO. E<p6crov 'tropa 'tO 'tE'tpanA.supo EMZN sivat syypa\jltJ.lO, ea tcrxf>ouv ot t<JO'tll'tE<; 'tOOV yrovtrov: M/ ENZ Kat £2 = MiN . ""

'

.....

sniKEV'tP11 'tll � a1t0 'tO t<JO<JKSA£� 'tpiyrovo '

=

EYKAEIAin:. A' 89 -r.l/37

""

""

MaO'IJUl'TlKoi A•aymvuYp.oi

------

An6 -ro tcrocrKsM� -rpiyrovo

IEM ,

sxouf.J£:

M, = E2

-------­

• A

A

An6 � -rps� 'tsAsmais� tcr&rrl-rs� yrovtrov, 1tp01Cfut'tSt 6n: ENZ = MZN Kat Ka'ta cruvsxsta, 'tO syyp<l'lftJ.lO -rs-rp<lnN::u po EMZN , sivat tcrocrKsM� -rp<l1tS�to. KE\rtpo -rou xsptysypaJ.tJ.lSvou K6t<Aou 'tO'U . tcrOcrKOOU� -rpan�iou, ea stVat 'tO <rr) f.l£t0 'tOJ.lll� . OUO 01totOVOll1tO'tS f.l£<JOKaes-rrov, OUO 0'1totovoilno-rs su&yp<lJ.lJ.lOOV 'tJ.l11J.l<l-rrov nou txouv <lKpa n� Kopu<ps� -rou. 'Emro A.om6v T -ro mtf.l£io 'tOJ.lll� 'tll� FK (1t01) sivat f.l£<JOKaes-ro� 'tll� EM ) Kat 'tll� f.l£<JOKaes-rou 'tll � EZ . L'tO -rpiyrovo -rropa FEZ 11 FT sivat OtXO'tOJ.lO� 'tll� yrovia� 'tOU Eiz Kat 11 or sivat f.l£<JOKaes-ro� 'tll� 1tAzup� 'tOU EZ . Apa 'tO <Jll!lSlO T ea avf]Ket cr'tO 1tSptysypa!l!lEYO K6t<Ao 'tOU 'tptyrovou FEZ .

1\u

E l<;

WV

QOK�

EWV tO U

tEUXOU<; 88

N12 .I:t Otoo,.tvo 9£TtKO aKtpaw N ypaq)(n>p£ ot;w 'TO\) 000 OUI(J)OpETlKcl Jlll J11]0EVlKcl \fiTI(J)la. 0 ap18J16� not> npmanna ow1ptiTa1 f.1£ Tov N Na npoGowpicJt:Tt: 'TI1 JJi'yum] O'Uvan] nJl'J] Tot> N. AUG11 Av yp<i'JIOUf.l£ os�ta -rou apteJ.loU N -ro Ot'Jiil<pto -rJ.lilJ.la ab , ').jyyro f.l£'ta-r6mmt� -rrov 'JI11<pirov -rou N Ka'ta ouo escrst� 1tp0� 'ta aptcr'tspa, 0 apte!l6� 1tOU 1tpOKU1t'tSt sivat o: •

M = Nab = 1 OON + l Oa + b .

'Exouf.J£: N M =:} N lOON+ lOa + b =:} N lOa + b = ab =:} N ::; ab . Exsiliil -ra 'J11l <pia a, b npsnst va

I

I

I

sivat Ota<popsnK<l f.l£'ta9} 'tOU�, 1tpoK61t'tst 6n 11 J.lS'YaAi>'tSp11 ouva-ril TIJ.lll 'tO'U

N sivat

98.

IIapa'tll pOuf.l£ 6n yta N = 98 , av npocrescrouf.J£ -ro ot'Jiil <pto 'tJ.lllJ.la ab = 98 AaJ.l�<lvouf.J£ -rov apteJ.16 M=9898=98� 10l nou Otatpsi'tat f.l£ 'tO 98. EnoJ.tSvro� 11 JlSytO''tll ouva-ril TIJ.lll 'tOU N sivat 98.

N13. Na npoGOlopiGt:Tt: 6Mn>� TO'U aKtpalO'U� n no'U £iva1 J..u<Jt� T1]� t�iGroG1J� =

(n - 1)(n - 3)(n - 5) · · · (n - 2011) n (n + 2) (n + 4) · · · (n + 2010) AUG11 AV 0

n

sivat apno�, 'tO'tS Kat 0 apteJ.16�

(

n n

+ 2 )( n + 4) . . . ( n + 20 10) sivat apno�, svro 0

( - 1) ( 3) ( 5) ( 2011) sivat nsptu6�, on6-rs osv sivat ouva-r6v va aA.11esust 11 3)( - 5) ( 201 1) = ( n + 2 )( + 4 ) ( + 2010) . s�icrromt ( - 1)( apteJ.16�

n

n-

n

n-

nn

· · ·

· · ·

n-

n-

n

n

· · ·

n

LS avaA.oyo crl>J.l1tSpacrJ.la Ka'taAllYOUf.l£, av unoescrOUJ.lS 6n 0 n sivat 1tSptu6�, 01tO'tS 11 napanavro s�icrromt osv sxst AU<rr) O''tO crUVOAo 't(!)V aKspairov.

A17. Na anoo£i�£T£ OTI 'Unapxo'Uv ant1pa Ttu1a Tt:Tpayrova 9tTIK<l>v aKtpa1rov Ta onoia Jlnopouv va napama9ouv ro� a9poiGfJ«Ta 'T11 � fJOp(J)'I]� 2° + 2m , ono'U n , m Ola(J)optTIKoi 9t:TIKOi aKtpaiOI. AuG1J IIapa'tll pOUf.l£ 6n yta n = m + 3 ' EXOUJ.lS 2n + r = 2m . 23 + 1) = 2m . 9 ' 01tO'tS yta m apno, SO''t(!) m

(

= 2k '* n ' AaJ.l�<lVOUf.l£:

n, m

)=(

n,

2n + r = 22k

(

•

9=

( 2k . 3 r .

EnoJ.tEvro� unapxouv anstpa �s011 :

2k) esnKrov aKspairov yta -ra onoia o apteJ.lo� 2n + 22k: sivat -rsl..sto -rs-rpayrovo.

I

ABCD t:"f'YPcl'JIIfJO 'TETpcl1tAE'UpO KQI M, L, N KQI K TU JltGa T(t)V nJ..t'U prov TO'U AB.BC, CD KQI DA ' aVTiGTmxa. Aivt:TQI OTI BMN MNC . Na anoo£i�£'T£ OTI: (a) DKL CLK . (p) TO Tttpcl1tJ..t'Upo ABCD tx�:1 tva �£'U'Yclpl MtvavTJ. nJ..t'U p� napclll1J�· OvKpavia 2010-11 r12. 'EGT(t) "

=

"

=

AUG11 (a) Enstoil

M, L, N Kat K -ra J.tScra -rrov nl..suprov -rou AB.BC, CD Kat DA , avricr-rmxa, sns-rat 6n KM 1 1 DB 11 LN Kat KN 1 1 A C 1 1 ML , on6-rs -ro -rs-rp<lnN::upo KMLN sivat napaM11A.6ypaJ.lJ.lO. EYKAEIAm A' 89 'T.l/38

------

MaOqJI.a'ttKoi AtayrovUYJI.Oi

-------­

ABCD sxoutJ.S 6n ABD = ACD , Apa sxoutJ.s: AMK = ABD = ACD = KND (1) A1t6 't11 crxscrTJ ( 1 ) Kat 'tl'JV U1t69tcrt') BMN = .Mile ' A.atJ.PavOUtJ.S KMN = 1 80° - AMK - BMN = 1 80° - KilD - .Mile = KilM .

(2) E1totJ.Svro<; 'tO 'tpiyrovo KMN sivat tcrocrKsM<; tJ.S KM = KN , o1t6'ts 'tO 'tt'tpa1tA.rupo K!vfLN sivat p6tJ.Po<;, o1t6'te Nf<L = NiK .

(3) OtJ.oiro<;, a1t6 n<; 1tapaU11A.is<; KM 11 DB 11 LN Kat KN 1 1 A c 1 1 !v1L , A.atJ.PavoutJ.S Kat 'tl'JV (4) DKN = Nic . tcr6'tl'J'ta yrovuov A1t6 n<; crxscrst<; (3) Kat (4) A.atJ.PavOUtJ.S DKL=DKN+ NKL=NiC+ NiK =ciK. (p) E1tstoiJ KM = KN , S1tt'tat 6n DB=2·KM=2·KN=AC. Ta 'tpiyrova ACS Kat BDS sxouv AC=DB, DBs = DiJA = ACD=ACs , ro<; syysypatJ.tJ.Sve<; cr'to ioto 't6l;o AD Kat I:xfltJ.a 1 BDS = 1 80° BDC = 1 80° - CAB = CAS .

E1totJ.Svro<; 'ta 'tpiyrova A CS Kat BDS sivat icra, 01tO'tS ea sxouv Kat AS = DS . Apa sivat SAD = SDA 1 80° - SAD = 1 80° - SDA DAB = CDA Kat a<pou ABC + CDA = 1 80° S1tt'tat 6n ABC + DAB = 1 80° BC 1 1 DA .

E1ttcrt')<; a1t6 'to syypcl\lfttJ.O 'tt'tpa1tA.rupo

...-

-

:::}

:::}

:::}

:::}

( y) 1tpaytJ.a'ttKffiv apt9tJ.rov x, y 1tou sivat A.ucrst<; 'tll <; El;icrrocrt')<; .Jx2 - 14x + 65 + Jl + lOy + 34 = 7 .

Al9. Av yta 'tOU<; 1tpaytJ.anKou<; apt9tJ.OU<; a,b,c,d tCJXl)st 6n a + b + c . + d = 1, b + c + d c + d + a d + a + b a + b +c , . c2 ' \+ , d2 a2 + b2 +� va ppshs 'tl'JV ntJ.iJ 'tl'J<; 1tapacr'tacrt')<; A = b + c + d c + d,+ � d .-t; a + b a \ b 4- c ) A20. Av a,b,c sivat esnKoi 1tpaytJ.a'ttKOi apt9tJ.oi, va a1tOOS{l;S'tS �t: aJ --+ bJ -cJ � ( a - b)2 + (b )2 + ( a )2 . Na S-,S'tacrt'tS 1t�'tS t St 11 tcrO'tl'J'ta.

--+ b+c c+a a+b ) Nl4. Na a1toos{l;s'ts 6n yta Ka9s q>UcrtK6 apt9tJ.6 n o apt9tJ.6<; f , n2 + 7n + 1 6 , , \ / , OSV Stvat aKspato<;. \ A( n) = ) ( n + 2 )( n + 3 )( n + 4 )( n + 5 ) - 24 Al 8. Bpshs 'ta �si>'Yll x,

---

.

\

.

, ( .., , � , ·.

-C

C-

,J=.

\

\

1.

1

,

·

·..

(

)

N l S. Bpshs 6A.a 'ta �� p, q 1tpcO'trov apt9tJ.cOV p, q tJ.S p

+

>

q yta

'tOU<; o1toiou<; Ot

apt9tJ.oi p q, p - q sivat Kat m ouo 1tpro'tot.

N 1 6. Av x, y sivat aKspatot ota<popsnKoi tJ.S'tal;u 'tOU<;, va a1toosil;s'ts 6n o apt9tJ.6<; B

�

+l � = x2 +x34xy -/ ---::-

--'--

r13. E>sropOUtJ.S 'tpiyrovo ABr . 'Ecr'tro 'tO tJ.Scro 'tll <; 1tArupcl<; AB Kat E 'tO tJ.Scro 'tll <; otatJ.Scrou v 'tO crT)tJ.tio Z sivat 'tO crutJ.tJ.S'tptK6 'tou B ro<; 1tpo<; 'tO E Kat 11 BE 'tStJ.VSt 'tl'JV 1tA.rupa Ar cr'to crTJtJ.Sio H, va a1toos{l;s'ts 6n BH HZ .

A�. A

=2·

EYKAEIAm A' 89 -r.l/39

'

Hpamffi1)� Xp1}«no�, r rEVtKO A1)K£l0 Pooot> Apy6p1) Davayuf>Ta, llpOTt>1tO llElp«Jl«TlKO rEV.A1)K£W Et>«TIUlK'I\� :EJ11)pV1)� A1t6 1toA:6 vrop� 'ta 1tat:xvt0u1 (61tro<; 'ta xapna tll <; 'tpa7touA.a<;, 'ta t;apm KA.1t) tixav 7tpoKa.Mcrtt 'to evotacptpov 1tOA.A.cbv JJ.aeru.ta'tlKcOV Kat J.l.t'ta9} aU'tcOV 'tOU Fermat, 'tOU Bemoulli, 'tOU Pascal K'tA. 1tOU 7tpOcr7ta6rJcrav va orocrouv JJ.ae11J.l.a'ttKt<; A.ucrtt<; cr' au'ta. LitJJ.tpa OJJ.ID<; tivat yvrom6, 6n fJ.OAt<; 'tOV 20°v atrova o Borel Kat mo 1toA.u o Neumann ( 1 928) Ka'tacptpav va tKcppacrouv fJ.t JJ.a6rJJJ.anK6 'tp67to tllV tvvota «1tatXViOt» . llo/..:6 1tp0xetpa ea J.l.1tOpoucrt va 1tet Kavt�, O'tt 'tO natxviot tivat 11 aVMUcrl') Kat 11 A.it '1'11 a7t6cpacrl')<; 1ttlV(J) crt JJ.ta unapK'tft npayJJ.anK6tll'ta 1tou yivt'talcrUyKPOUCJll (J'I)JJ.q>tp6V'trov.

Ta natxviOta JJ.t Kpt'tftpta tllV tcpapJJ.orft JJ.tcra <JtllV Ka6rJJJ.tptv6tr�'ta ,xropit;oV'tat crt 'tptt<; Katr�ropie<; : y) Kotvro\'tKa natxv£ota a ) L'tpatr�ytKa Kat TioA.mKa natxviota Jl) OtKovoJJ.tKa natxviota Ta 7tpro'ta avanwxe11Kav Kat JJ.tA.e'tft6rJKav KUpiro<; cr'to B ' lla-yK6crJJ.to n6A.eJJ.o, 'ta oei>'ttpa fJ.t tllV avrurro�11 'tOW 7toA.ut6vtKrov t7ttXttpitcrtrov Kat 'ta 'tpha fJ.t tllV t7ttcr'tftf.l11 tll <; KotvrovtoA.oyia<; . TtA.o<;, avaA.oya, av ot 1taiK'tt<; fJ.1tOpouv va t7ttJJ.�aivouv it va t�ap-ra'tat 'tO natxviot a7toKA.etmtKa a1t' tllV rum, txouJJ.t a) 'ta nat:xvtOta mpatr�rtKit<; Jl) 'ta natxviOta rum<; Ta natxv£ota mpatr�ytKft<; 7tapoumat;ouv J.l.t'YMo evota<ptpov Kat yt' au't6 CJtll (J'I)V£xtta ea acrxoA.11eOUJ.l.t J.l.OVO 11' au'ta. AKOJ.l.11 unaPXOUV natxviOta 1tOU cpaivov'tat aeroa Kat oiKata, evro oev tivat. Le 'te'tote<; ntpl1t'tcOCJet<; 11 etropia natxvtotrov JJ.a<; naptxtt optcrJ,J.tvou<; 'tp6nou<; va cpuA.ay6JJ.acr'tt. fiiH;(VlOlU OUO UTilJ.WlV

Luvftero<;, cr' au'ta 'ta natxviota, ot naiK'tt<; 11nopouv va t7ttfJ.�aivouv Kat 'ttAtKa 6n xavtt o tva<; naiKtll <; va 'tO Kepoit;tt 0 MAo<;. ft' au't6 'tO A.Oyo 'ta natxviOta aU'ttl ovofJ.tll;oV'tat 1tWXVi�za �60 1tWK'rWV J-18 pTf&Vu'o aOpozupa. llaV'tro<; cr' au'ta 'ta natxv£ota u1taPXouv an' tllV aPXft optcrJ,ltvot KavovtcrJJ.oi (Kav6vt<;), 1tou 7tptntt Kat ot ouo 1taiK'tt<; va tll POi>v cr' 6A.11 tll otapKtta .'tou 7tat;{Vt0tou. Ot Kav6ve<; au'toi it Kaeopit;oV'tat M ' 'tOU<; 2 naiK'te<; it U7ttlPXOW ava-yKamtKa an' tll q>Ucrl') 'tOU 7tatxvtotou JJ.tcra cr'to cruyKtKptJ,ltvo XIDPOXP6vo 'tOU. Ot ouva'tOtll'te<; tntJJ.�acrtro<; nou txouv ot ouo naix'tt<; 'tou<; ovofJ.tit;oV'tat cr'tpa'tl')ytKS<; it 'taKnKt<;.

A<; unoetcrouJJ.t 6n txouJJ.t ouo naiK'tt<; A Kat B. Eivat }.oytK6 o KnOt: naiK'tTI� va t:<pappoot:t pta GTpaTI)yuo) Tnota roCJTt: va peyt.CJTonot'l)ot:t TO Ktp()o� TOl> 1\ CJTI)V nt:pin'troCJTI no'U t:iyat a3uvaTo va' Xt:t Ktp3f1, va t:AaXtCJT01tOt1\CJt:t Tl) �flf1Ul TO'U. Ot mpatr�ytKt<; 'trov natK'trov (o11A.. 'tO ffilvoA.o 'trov Ktvftcrtrov Kaet naiKtll ) 11nopouv va napacr'taeouv fJ.t tvav . mvaKa nou OVOJ.l.Ust'tat « mvaKa<; nA.11proJJ.c0v ». 'Eva<; 'tt'toto<; mvaKa<; eivat 61tro<; napaKU'tffi :

AI A2

B2 f12 f22

BI fu f21

Bv f1v f2v

... ... ...

:

A, .... f12 fKY fKI Ot ypafJ.f.l£<; AI> A2 , , AK 'tOU nivaKa eivat ot cr'tpatr�ytKt<; 'tOU A Kat ot cr'tftA.e<; BI> B2 , , Bv eivat ot mpatr�ytK£<; 'tOU B. Ot apteJJ.oi fu, . . . , fKV eivat eenKoi it apVl')'tlKOi. L'tOV nivaKa O'tl tivat apVl')'tlKO 'tO xavtt 0 A, evro 6n tivat eenK6 'tO Kepoit;tt 0 A it 'tO xavtt 0 B. H aVMUcrl') Knee 1tat;{Vt0tou ea yive-rat U7t ' 'tOV mvaKa Ktpoou<; 'tOU. 'E'tm 1tX· cr'tO yvrom6 1tatxvilit 1ti:rpa-1f!oli�z-xapr:i 0 mvaKa<; 'tOU nat;{VtOtou yta ouo naiK'tt<; A Kat B tivat: . . •

Tu

Ka9ap6: 7tatxviota

. • •

B1

A 1 (1te-rpa) A2 ('I'Mi<it) A3 (xapn)

(7te-r pa)

0 -1 1

B2 (waA.wt) 1 0 -1

B3 (xapn) -

1

1 0

LUJ.l.q>roVa fJ.t ama 1tOU txouJJ.t 1ttl J.ltxpt 'tcOpa, tivat <pavtp6 6n 0 crKonO<; 'tOU A eivat va f.lt'YW't01t0titcret 'tO KtpOO<; 'tOU, evro 'tOU B va eAaXt<J't01tOtitcret 'tO KtpOo<; 'tOU A. 'E'tm 0 A yvropit;et 6n av 0 B tivat £9mvo<; naiKtll<; ea 'tOU tm'tp�et va napet 'tO mo A.iyo 1tOU yive-rat. �auou 0 A cr'tOV mvaKa nA11po>f.lcOv 'tOU Ota/J:yet ypaJJ.fJ.t<;, evro 0 B cr'tfJA.e<;, apa 'tO eAftXJ.cr'tO Kaf}e ypallJ.l.it<; ea 'tOU tm'tpt7tet 0 B va KepOit;et. fta 7tapaottyfJ.a, a<; etropitcrouJJ.t tva 7tatXViOt fJ.t 'tov t�it<; nivaKa Ktpoou<;:

AI A2 A3

At

max

Bl

4

3 1 5 5

B2 7 8 1 1 8

B3

min

2 2 1

2 2 1

2

2

-4

EYKAEIAID; A' 89 T.l/40

-4

------

H 9£mpia Ilatxv..Oui>v Kat 'T« Ma&qpanKO.

-------

01t6-tt av 1t.x. o A 1ta�tt 'TTIV A2 mpa'TTI'Ylrit 'tOt> (OTJA. 'Tl1 211 ypaJ.l.J.l.TJ), 't6'tt o B Oa 'tou em'tp£'1ftt va 1tnptt 'tO eAaXt<no amijc; 'Tile; ypaJ.l.J.l.TJc;, 1tOU tivat 'tO 2 (OTJA. o B 9a 1ta�tt 'TTIV B3 <npa'TllytJCit 'tOU) . Avn<nOtXa, o B yvropll;tt, 6'tt 0 A ea 1tpOcr1ta9ti va 'tOU 1tnptt 'tO 1teptoo6'ttpo ouva't6 Kepooc;. C'tot av 1t.x. 0 B 1ta�tt 'TTIV B2 <npa'TTI'Ytrit 'tOU (OTJA. 'Tl1 211 <nl'JA.TJ) 't6'tt o A J.11tOpei va 'tOU 7tnptt 'tO �<no 'Til e; <niJ A.TJc; au'tijc;, 'tO 8.

TtAtKn, a1t' 6A.a 'ta tAaXtO'ta (min) 'tO)V ypaJ.l.J.l.OlV 'tOU, 0 A ea 1tpOcr1taeijott va 1tnptl 'tO �t<no, evro 0 B a1t6 6A.a 'ta �t<na 'tO>V O'tTJAroV 'tou, va xnott 'tO tA«lXt<nO. Av au'toi ot ouo apt9J.l.oi tivat ioOt ( min(max) = max(min) ), 't6'tt AeJ.l.t 6n txouJ.l.t tva Ka9ap6 1tatxviot (von Neumann ' s Minimax Theorem). �'to 1tapaott'YJ.l.n !lac; txouJ.l.t tva Ka9ap6 1tatxviot ot6'tt min(max) = 2 = max(min) Kat OUJ.l.<peptt o'touc; 1taix'ttc; A Kat B yta va J.l.TJV txouv 1ttpt1te'tttec; va OUJ.l.<provijoouv <nov ouvouaoJ.16 (A2, B3) iJ o'tov (A�.

2 B3), ot6'tl e'tot J.16Vo o A Ktpoll;tt oiyoupa, ev ro o B xnvtt 1 -3 0 AI 6oo 1tto A.iyo yivt'tat (oTJA.. 2 XPTJJ.l.anKtc; J.l.Ovnotc;). 'Eva 3 4 A2 aK6J.l.TJ 1tapaott'YJ.l.a Ka9apou 1tatxvtotou txouJ.l.t J.1t 'tov t�l'Jc; mvaKa 1tATJproJ.l.roV: 4 0 A3 E1taATJ9tU<nt 'tO !

1 1 �

B1 22

Ot JU:tK'Tt� G'TpU'Tfi'Yt.Ki�

B

B3 2

D'tav <J'tOV 1tiVaKa 1tATJpro!lffiv l1 <npa'TllytJCit 'tOU KPt'Tll piou minimax oev J.l.1t0pti va oroott aJ.l.tytic; O'tpa'TTI'YtKec; (Jlt aAA.a A6yta 6'tav 'tO �t<no 'trov tA.aximrov 'trov ypaJ.l.J.l.OOV eivat otn<popo 'tOU tA.aximou 'trov O'tTJAroV), 't6'tt AeJ.l.t 6-rt £xoUJ.l.t tva 1tatxv{ot J.l.tK'tij c; O'tpa'TllytJCitc;. �'TTIV 1ttpimro0TJ 'Tile; J.l.tK'tl'J c; <npa'TllytJCitc;, oev J.l.1tOpOUJ.l.t Jlt �t�at6'TTI'ta va OUJ.11ttpnvOUJ.l.t Ot 'tl 1tp£1ttt va OUJ.l.<provijoouv Ot A, B Ot6'tl oev U1tnPXtt Kavtva OTJJ.l.tio tooppo1tiac;. M1topOUJ.l.t 6J.1COc; va J.l.tAnJ.l.t yta 1tooomn tmroxiac;.

E>tOOPTJJ.l.a (Neumann): 0 1taiK'Tll c; A txtt J.l.ta 'tOUAaXt<nOV npt<TtTJ J.l.tK'tij <npa'TllytJCit 1tOU av 'TTIV aKOAoUeijott 'tO J.1ioo Ktpooc; 'tou eivat Jlt'YaAU'ttpo iJ ioo 'tou V 61tou V l1 'ttJ.l.TJ 'tou 1tatXVtotou. AV'timOtxa, u1tnPXtt J.l.ta 'tOUAaXto'tov npt<TtTJ J.l.tK'tiJ <npa'TTI'Ytrit yta 'tOV B 1tOU av 'TTIV aKoA.ou9Tjott l1 J.1iOTJ �TJJ.l.W 'tOU tivat J.l.tKp6'ttPll iJ lOTJ 'tOU V. IYJpeiwaYJ: �'ta Ka9apn 1tatxviota V = 'to OTJJ.l.tio tooppomac;. EmOTJc;, ma Ka9apn 1tatxviota J.l.1topouJ.l.t va �pouJ.l.t 'to OTJJ.l.tio tooppo1tiac; Kat J.1t J.l.ta MAll J.1i9ooo, 'Tl1 J.1i9ooo 'trov u1toote<ntprov mpa'TllytKrov. �'TTI J.1i9ooo a.u'tij ota.ypn<pOUJ.l.t 6oo 1ttptoo6'ttptc; u1toote<ntptc; <npa.'TTI'YtKec; ro<nt o 1tiva.Kac; 1tAllProJ.l.roV va.' vat o a.1tA.oumtpoc; ouva.'t6c;. Mta. 'YPClJ.l.J.l.TJ (yta. 'tOV 1tCllK'TTI A) a. eiva.t U1t00tt<ntPll J.l.tac; MAllc; 'YPClJ.l.J.l.TJc; � 6'ta.V KMt <JTOlXtio 'Tile; a. tivat J.l.tKp6'ttpo iJ ioo a.1t6 KnElt <notxtio 'Tile; �. Av'timOtxa., J.l.ta. <nl'JA.ll (yta. 'tov 1ta.iK'T11 B) a. eiva.t U1toot£<ntPll !lta.c; «lAA11 c; <nl'JA.llc; � 6'ta.v KnElt o'tOtXeio 'Tile; a. tiva.t Jlt'YnAU'ttpo iJ ioo a.1t6 KMt <nOtXtio 'Tile; �. H J.l.eeoooc; 'trov u1tooe£mtprov <npCl'TTI'YtKOOV OOUAtUtt yta.'ti Ot U1tOOte<ntptc; <nPCl'TTI'YtKec; 1tO'te Oev 9a. t<pClpJ.16�oV'ta.t a.1t6 tva.v op9oA.oyt<Ttij 1tCllK'TTI (!l' aAA.a A6yta. £xouv J.l.TJOtvtJCit m9a.v6'TTI'ta. t<pa.pJ.l.oyl'Jc;). �'ttc; JlttK'tec; O'tPCl'TTI'YtK£c; 9a. t<pa.pJ.16�0UJ.l.t 1tnV'ta. 'T11 J.1i9ooo 'troV U1t00te<ntprov <TtPCl'TTI'YtKOOV Kat ouvij9roc; KCl'taATJ'YOUJ.l.t o' tva.v mva.Ka. 1tA.11pro!lffiv 2x2 . n.x. mov ...:.. ·I-:.

A 1 ---+ · ___::: ---1 ·-=----+ T6'tt ouvtxil;Ot>Jlt roe; t�Tjc; : A2 OvoJ.l.�OUJ.l.t X 'TTIV mea.v6'TTI'ta 0 1tCliK'Tll c; A va ClKOAoueijott 'Tl1 L---"------'---'----L-------' <npa.'TTI'Ytrit 'tOt> A�. 'tO'tt 1 -x tiva.t TJ m9a.v6'TTI'ta. va. a.KoA.oueijott 'Tl1 A2.

1-y) (:>. (y) _--i .::... _.L Ji.. ::.. _ .....::l..

OJ.l.oiroc;, t<nro y 11 mea.v6'TTI'tCl 0 B va. a.KoA.ou9Tjott 'Tl1 <nPCl'TTI'Ytrit 'tOU A-"t-" (� · -----+-------1 x)'----+ 1-B3, 't6'tt 1 -y tiva.t 11 1tt9a.v6'TTI'ta. va. a.KoA.oueijott 'Tl1 <npa.'TTI'Ytrit 'tOt> B4. . .::.:: ....:x .::.):.__..L_-=.; {=1.: � 'E'tot , yta. 'tov 1ta.iK'T11 A £xouJ.l.t 6n : LA2 ---l.-=..---__.

V(A, B3) = 2x - 5(1-x) = 7x - 5 Kat V(A, B4) = -2x +0( 1-x) = -2x E>t'toV'tac; V(A, B3)=V(A, B4) OTJAaOTJ 7x-5=-2x �pioKOUJ.l.t 6-rt x=5/9, npa. 1 -x=4/9.

OJ.l.oicoc;, yta. 'tOV 1ta.iK'T11 B £xouJ.l.t 6n : V(B, A 1) = 2y - 2(1 -y) = 4y - 2 Kat V(B, A2) = -5y +0(1-y) = -5y et'tov'ta.c; V(B, A1) = V(B, A2) o11A.a.oiJ 4y-2=-5 y �piOKouJ.l.t 6-n y=2/9, npa. 1-y=7/9.

Tropa., 11 'tlJ.l.TJ 'tot> 1ta.tyviou �piOKe-rat Jlt a.vnKa.'tn<na.OTJ 'trov 1tt9a.vo'tij'trov (x iJ y) ot o1tota.Oij1tO'tt a.1t ' 'ta. V(A, B1) iJ V (A, B2) iJ V(B, A1) iJ V(B, A2), OTJAaOTJ tiva.t V=-1 0/9.

To <pt>otK6 v6TJJ.l.Cl 'Tile; 'ttJ.l.l'Jc; 'tOt> 1ta.tyviou eivat 6'tt, t<p6oov t1ta.VaAll<p9ei 1toAA.tc; <poptc; 'tO 1tatXViOt, 'tO J.1ioo (a.va.J.l.evoJ.l.tvo) Ktpooc; 'tou A ot �npoc; 'tou B roe; 1tpoc; 'to JltpiOto a.yopnc; tivat -1 0/9 (oTJA.a.OiJ -1 , 1 1 %) 1tou 1tpa.KnKn OTJJ.l.CllVtt O'tl J.l.ClKp01tp69tOJ.l.Cl 0 A xnvtt 1 ' 1 1 1tOOO<nta.iec; J.l.OVnotc; 'ttc; 01toiec; Ktpoti;tt 0 B. IYJpeiWaYJ 1: H AUOTJ 1[0\) xeptypn'lfClJ.l.t 1ta.pa.1tnvro t<pClpJ.16�t'ta.t 1tnV'tO'tt 6'ta.v etA.oUJ.l.t va. �pOUJ.l.t tOa.VtKic; J.l.tK'tec; mpa.'TllytKec; ot 1tatxviota. 2x2.

IYJpeiwaYJ 2: To OUJ.l.1tepa.oJ.l.a. yta. 'TTIV 1ta.pa.1tnvro a.v«lAuOTJ tiva.t 'tO �ijc; : H cr'tpCl'TTI'Ytrit (5/9, 4/9) tiva.t toa.vtrit yta. xov 1ta.iK'T11 A Ka.t ll <npa.'TTI'Ytrit (2/9, 7/9) tivat t3a.vtJCit yta. 'tOV 1ta.iK'TTI B. M' aAA.a A6yta. o 1ta.ilc'T11 c; A 1tp£1ttt va. t<pa.pJ.16�tt 5 <poptc; 'TTIV mpa'TTI'Ytrit A 1 Kat 4 <poptc; 'Tl1 mpa.'TTI'Ylrit A2 evro o 1ta.iK'T11 c; B 1tpexet va. t<pa.pJ.16�tt 2 <p6ptc; 'Tl1 o'tpa.'TTI'Ytrit B3 Kat 7 <poptc; 'Tl1 mpa.'TTI'Ylrit B4 KnElt 9 <poptc; 1tou 1ta.ll;e-ra.t 'to 1tatXViOt a.u't6.

IYJpeiw(!YJ 3: 'Eva. 1tatXViOt J.1t 'ttJ.l.TJ J.l.llOtv oVOJ.l.�t'ta.t oiKa.to.

IYJpeiwaYJ 4: 'Eva. 1tatXVWt OVOJ.l.n�e-ra.t OUJ.l.Jlt'tptK6 6'ta.V 0 avn<nOtXOc; 1ttVa.Kac; 'tOU tivat OUJ.l.Jlt'tptK6c;, O'ta.v OTJA. ypa.J.l.�c; = <nl'JA.ec; Kat ClJ.l.Ot�a.ia. . evaA.Aa'YiJ a.u'troV oivtt 'tOV tOto mva.Ka.. llx. 'tO 1tCltXVWt 7Ci:rpa-11faJ..ioz-xap-ri tiva.t OUJ.l.Jlt'tptK6 Kat npa. oiKa.to Ot6'tt KnElt OUJ.l.Jlt'tptK6 1tatxv{Ot eiva.t oiKato.

IYJpeiwaYJ 5: duo OtclOTJJ.l.Cl 1tatXViota. 2 a.'t6J.l.rov tivat a.) To L1iA.1JJlf.la -rov Kpa-rovJlevov Kat �) H Ma:m -rwv <PVA.A.a>v.

l I _25B3 I B2 -o 4 1l-------:- --+..:_B32'-": 5'-'""-'----f ·B� I I�

_ _

EYKAEIAIII: A' 89 'T.l/41

MoOolvw v' oyonw TO MoOrJJIOT I Ka Aycnrq tlva 1 va

Jlll

<popaoa 1 auTo nou ayanofi

======

1 . l:;oKOAU'TE� , L 'EVU� EtXE • crOKOAu:tE� .

, llOUAllcrE .

'tU

ruopyo� MEV�rovi01)�

2

7

a1t6 auTs�. L\roptcrs T� 30Kat Tou EJ..tEt­ vav 20 . TI6crE� Eixs cr'tllv apxfl; 2. nooo<J<patpo ·� I:To 1tpona9A.11J.La 1tOOocr<paipou. '01tro� sivat yvrocrT6, Ka9s OJ.LaOa 1taipvst 3 pa9J.Lou� yta 't1l viKll 1 pa9J.L6 yta 'tllV t­ cro7tal..ia Kat Kavava pa9J.L6 yta 'tllV ftna.

Mta OJ.LaOa as 20 ayrovs� sixs Ot1t'Ailcrw� viKE� a1t6 Tt� ftns� Kat ouo tcro7taJ..iE�.

TI6crou� pa9J.Lou� cruyKEv'tprocrE au'tft 11 OJ.laOa cruvol..tKa;

�� �·

4. Ta (J'T1)AO

3. Xp1>m] aJ..1><Jioa

'Eva� Ta�tOtro'tll � sixs J.Lta XPUcrft aJ..ucrioa J.lE 7 KpiKou�. I:uJ.L<pro­ VllcrE J.lE To �evoooxo Tou va Tou osiV11 Ka9s ppaou a1t6 ava Kpi­ Ko. fta'tt apKEl VU K6\j1Et Kat VU <l1tOKOMTJcrEt 'tOV 'tpt'tO KptKO; 'Eva� ay6paas crroJ..O M1tM, K6KKtva Kat Maupa.

'OJ..a ftTav M1tM EK'to� a1t6 ouo, 6J..a ftTav KoKKtva EK'to� a1t6 ouo Kat 6J..a ft'tav J.Laupa EK'to� a1t6 ouo; ll6cra crroJ..O ay6paas;

5. To <JK01>AtKaKt

'Eva crK:ouJ..11KaKt TproyovTa� To xapTi avoiyEt op6J.Lo J.LEcra a1t6 J.Lta E�aTOJ.lll syKUdo7taiosta .

8EKtVaEt U1t0 'tO J.l1tpocrnv6 E�cO<pUMO 'tOU 1tpcO'tOU 'tOJ.lOU Kat crE EU9Eia YPUJ.LJ.lit KU'tal..ftyEt cr'to mcrro E�cO<pUMO 'tOU EK'tOU 'tOJ.lOU. Ot 'tOJ.lOt ElVat To1to9E'tll J.lEvOt cr'tll PtPJ..w9ftK11 J.lE 't1l crstpa a1t6 Ta aptcrTEpci 1tpo� To os­ �ta. Av Ka9s TOJ.lO� sxst 1taxo� 5 EKaTocrTa, 1t6cro sivm To J.LftKo� 'tll � ota­ opoJ.Lit� 'tOU crKOUAllKtOU;

6.

T a fJ1t01>Kal..ta

I:E Eva crOU1tEp J.LapKE't yta Ka9E 5 yua.J..tva aOEta J.l1tOUKaJ..ta UVU\jiUKnKou 1tou E1tEcr'tpE\jlav E1tatpvav ava YEJ.lci'to.

TI6cra J.l1tOUKal..td ava­ \jiUKnKou 9a 1ttti J.Lta otKoyEvEta 1tOU 01.YyK:E­ V'tprocrE 28 oosta J.l1tOU­ Kal..ta; EYKAEIAHI: A' 89 T.l/42

------

7. Ot 1tt'TGE<;

Mu9aivro v' ayanro -ra Ma8qpa-r1Ka

-------

�E Jll<l Jl<l9TJnldJ EK�poJ.Li], o Bayyti..TJ <; eixe 3 JltKpE<; 1ttTCJE<; Kat o ftffipyo<; �TTJV 1t<lpEa 1tpOCJTE9TJK€ 0 �llJlllTPTJ<;, 0 01tOtO<; CJTO TEAo<; TOU <j><l'YTJTOU 1tAi]pcocrE CJTOU<; �UO UAAoU<; TO JlEpi�t6 TOU 1tOU i]TaV . II6cra wpffi (l1t0 ama ea 1tUp€t 0 Bayyti..TJ <; Kat 1t0(j(l o ftffipyo<;;

5.

8€

'Eva<; Jl<lVU�TJ<; EXEt CJTO Jl<l'Y<lsi TOU JlUl suyapt<l Jl€ TECJCJEpa suyt<l KtA..fV fi . (crT<l9J.L<l) TCOV 1, 3, Kat To1to9ETOOVT<l<; T<l si>yta CJ€ CJUV�U<lCJJlOU<; CJTOU<; �icrKOU<; TTJ<; suya­ pt<l<; Jl1t0p€i va s'U'YtCJEt €Jl1tOpEUJl<lT<l ax6 1 ECO<; 40 Ktl..a ;

9

27

9. To E'Upro

27 + 2 29€ =

Ot TpEt<; n<; xapta<; t<payav i]mav Kat �taCJKE­ �acrav cre J.Lta Ta�tpva.

0 A.oyaptacrJ.L6<; i]Tav Kat o Ka9tva<; t�cocre crTo yKapcr6vt ax6 10€. Ax6 T<l xou TOU<; extcrTpE\jiE, t�cocrav crTo yKapcr6vt Kat o Ka9tva<; Tou<; xi]pe ax6 tva eupffi. Apa Kat ot Kat CJTO yKapcr6vt, CJUVOAtKU TpEi<; E�COCJ<lV 3 = I Ilou xi]ye TO tva eupffi; ·

25€ 2€ 9 27€

5€

2€

1 0. l:OKOA«'TclKl« K«l J10A.-6pta 'ExouJl€ J.Lta craKouA.a Jl€ croKoA.aT<iKta Kat Jlta craKouA.a JlE JlOAU�ta.

Ilco<; Eivat �UV<lTOV va �pOUJl€ 1t0t(l CJ<lKOUA<l EXEt T<l 1t€ptcrCJOTEpa xcopi<; va T<l JlETpi]crOUJl€ Kat xcopi<; V<l T<l S'U'YlCJOUJ.LE;

EYKAEIAHI: A' 89 -r.l/43

To 1° rYMNAI:IO I:KAAAI: !lP!lllOY S1tavspxstat JlS 211 E K9 E J: H MA9 H MATI Kn N

:Eto 1° te6xoc; tllc; 1tporryouJlSVll c; XPOVuxc; tou 1ttpto3u<Ou «Eurl.si3TJc; A '» 1tapoucruicraJ.lS tllV 111 SK9S()TJ f.l<l9TJJ.l<ltl.KOOV tO'I) 1 ou fUJ.lV :EKciA.ac; !lpro1tOU. :Eto 1° tsuxoc; tllc; <pststvf)c; crxoA.tld]c; XPOVtcic;, EXOUJ.lS t1l xapa va 1tapoumcicroUJ.lS tllV 211 SK9S()TJ f.l<l9TJf.l<ltlKOOV tO'I) tbtOU crxoA.siou.

'OA.TJ tllV 1tSptcrtvi) XPOVtci, 01. f.l<l9TJtEc; K<lt 01. f.l<l9'Jltptsc; trov tf.lTJf.lcltffiV A., A2, A3, At, r 1 Kat r2 JlS tllV Ka9TJrfltptci touc; trov f.l<l9TJJ.l<ltl.KOOV K MapaYKOV n-:ropyia, StOlf.l<l<J<lV xropic; va xacrouv ropsc; f.l<l9TJJ.1cltffiV, yta clMTJ f.ll<l <popci 1tprot6ro1ta Kat Sf.11tVSUcrf.!Sva <l1t0 t<l f.l<l9TJf.l<ltl.Kcl spya J.1S titA.o: ' ' <JTaM:c; Ma91]J1aTtK<f>v <JTO vou, 1tA1]J1J1'6pa ropaiot> <JTl]V 'lll>Xil '' 'Epya 1tou f.11t6pscrav va 3rocrouv crta f.l<X9TJf.l<lttKci J.lta 3ta<popsttld] Kat sv3ta<pspoucra 3tcicrta<JTJ cruvMovtcic; ta JlS 1toA.A.ouc; toJ.!Sic; trov s1tt<:rtllf.100V Kat trov tsx;vrov, JlS tllV apxttSKtOvtld] Kat t1l XTJJ.lsl<l <lMcl Kat JlS t1l f.lOUcrtld], t1l sroypa<ptld], tllV �UAoyAU1tttld], tllV xaptOK01tttld] Kat t1l xapaKttld]. H crUVbS<:rTJ autit S1ttte6X9TJKs f.lS tp01t0 bS�totSXVlKO Kat aptcrtOtSx;vtKO. fta V<l K<lt<l()l(SUcl<JOUV t<l spya touc; 01. f.l<l9TJtEc;, clVtATJ<:r<lV tbssc; <l1t0 ttc; 3tci<popsc; sv6tlltsc; trov f.1U9TJf.l<lttKrov 1tou 3t3cix9TJK<XV tllV 1ttpacrf.!SvTJ XPOVtci <XMci Kat a1t6 ne; ysvtK6tspsc; crusTJtit<:rstc; 1t0'1) SK<lV<lV yta ta f.l<l9TJf.l<ltl.Kcl <JtllV tcl�TJ f.lS tllV K<l9TJrfltpta touc;. :Exs3iacrav Kat KatacrKe6acrav ta 1tA.atrovtKci Kat ta apxtf.litbsta crtspsci, ta crtspsci sK 1ttptcrtpo<pitc;, 1tpt<:rf.1ata, avtt1tptcrf.1ata, 1tUp<XJ.1i3sc;, ap9protci crxflf.lata Kat 1toA.A.a aA.A.a f.1S «SUtsA.it» UAl.Kcl xaptta Kat xapt6Kap<pa, KaAaf.lclKt<l it 1tA<l<JttK6.

0 S1tt()l(S1ttllc; tllc; SK9t()TJc; tvro1trocrtas6tav OXl f.lOVO <l1t0 tOV 1tAoUtO trov SK9Sf.1cltffiV aMci Kat <l1t0 tllV U'I'TJAit atcr9TJttril trov spyrov. Ot f.l<l9TJtEc;, a<pou S1t1t0'1) spycicrtllK<lV <ltOf.ll.Kcl it <JUMQ'Yl.Kcl, a<pou 1tprota JlSAEtll<J<lV to 9Sf.l<l touc; Kat E'YP<l'l'<lV t(l crxsttKcl KStf.lSV<l <Jtll cruvsxsta crxs3iacrav Kat K<lt<l()l(eU(l<J(lV JlS 1tOAU KS<pt Kat JlSPclKt t(l SK9Sf.1<lt<l, �cisovtac; tO 1tpocrro1ttK6 touc; youcrto. 'Etm to i3to 9sf.1a, yta 1tapci3styf.1a to K6crKtvo tou Epatocr96vTJ, to

EYKAEIAm A' 89 T.l/44

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

2 '� 'EK0&GTI MaOqp.o:nKrov-----·-------

epM:m::c; aTTJV SK9E<JT) va EtVat Ka'taaKwaa�o 1tclV(I) aE xapri it as <pullo XaAKOU. Ta EK9sJ.taTa ava<pspov-rav : l:TTJ yewfJ.. epfl.'f/Vela Tr/� G1ClJJepunuaj� u5z6rrJra� rov noYfJ.ov w� npo� rrJV np6aOeurJ Kaz rrJV arpaipeurJ, rrJV aV1:lJ.leTa8£TlKij u5z6ritm r17� np6a0ea'f/� Kaz rov noYfJ.ov, -ra JJaylKa -reTpaywva, ro KOtTKlVO rov Epa-rouOtv,, -rove; -rpiywvov� Kaz rov� -r£Tpaywvov� apz8fJ.OV�, m KlauJJa-ra, rz� hvvaJJel� rwv rpvazKdJv apzOfJ.dJv, rov� ti.PaKec;, m aplOJJ'If'lKa uVJJPola an6 rrJV apxazorrJm, TrJ f1,i8obo vnoA.oyzaf..lOV rov a8poiaf1,ro� 1 + 2+. . . . 99 an6 rov Gauss, TYJV mzvia rov Mobius, ra ap8pw-ra qlj}laf'a {KVpra - /11'/ KVpra), TOV KVKlO Kat TOV KVKllKO hitTKO, 'rl� yrovit:c;, TrJ yeWfJ.. epfJ.'f/Veia Tr/� ravr6rrJm� (a+PF = a2+2ap+p2, ro -rpiywvo rov Pascal, -ra 1Cla-rwvlKa Kat apxlJJijbela u-repea, m mepea eK 1C£plt1f'pO(/Jij�, 'tTJ uxtu, O'fKWV KVlivbpov Kat KWVOV 1COV exovv ibza flaa., Kal ro ibzo Vlj/0�, TrJ crxs<JT) O'fKWV 1Cpiu}laf'O� Kal 1CVpa}liba� 1CO V exovv flaaez� iaa rpiywva Kal iaa Vlfl'f/, 'r'f/ «fJ.aO'f/fJ.ariK� rexv'f/» rov ESCHER, TOV M,xavltT}lO� f'WV AVf'lKVOijpwv.

AVclJ.tEaa a-ra EK9SJ.laTa TTJc; SK9E<JT)c; 0 E1ttcrKS1tTTJ<; Jl1tOpouaE va ppst TO spyo TTJc; 1tpOKOJ.lJ.Iivtlc; JlSAtaaac; aMci Kat TTJV E�itrrl<Til yta'tt av'tft Va <pTtclXVEt JlE TO KEpt TTJ<; KVpit9psc; ax6 KaVOvtKcl E�ciyrova Kat OXt TE'tpciyrova it ta6xi..E'U p a Tpiyrova. Yxftpxav KpUaTaAAot NaCl Kat ciMrov X'lJ.llKIDV evroasrov 1tOV TJ KpvaTaAtKft OOJ.lit -rove; Eivat yvroa-rci YEffiJ.lETptKci aTEpEci KavovtKci Kat JlTJ. Yxftpxav aK6Jla Kat -ra ysroJ.tE-rptKci crxflJ.ta-ra J.1E ne; 6J.1op<psc; crl>JlJ.lETpiEc; xov 1tETclJ.lE 0 evac; aTOV ciMov 6-rav xaH,;ovJ.tE XtOV01tOAtJ.10 Kat OEV TO cr1>VEtOTJT01tOtOUJ.lE, vxftpxav P6-raaA.a JlE ne; xapciMTJAtc; Kat ne; TEJ.lVOJ.tEVEc; wesiEc; 1tOV <pncixvst oit9EV J.taytKci TJ <pU<JT) 1tclV(I) -rove;. OA.' av-rci aMci Kat ciMa xoUci oev it9tM: va xapaA.tt\jiEt va -ra J.taPTVpitatt a-rove; J.ta9TJTBc; TTJc; TJ K Mapa-yKou. Exstoit -ra J.ta9TJJ.lanKci axooioovv xta-rci -rove; Kav6vsc; TTJc; <pu<JT)c; o t1ttcrKS1tTTJ<; ata9av6-rav va xA.avciTat y'\)pro TOV aTTJV sK9t<JT) TO tpiDTTJJ.ta: « Etu:i� Kat TJ <puaq. Y CJTEpoupE 1] xpOt)yoi>paGTE ; »

l:-ra E-yKaivta TTJc; SK9t<JT)c; SKT6c; ax6 -rove; Ka9TJyr)TSc; Kat TOV OlE'U9vv'tft TOV axoA..eiov, xapsppsOTJKav o oitJ.tapxoc; !lproxirov K Ot.KovopaKo� TJ l:xoA. l:uJ.tPovA.oc; -rrov Mq,9TJJ.tanKffiv TTJc; xsptoxftc; Kat xoUoi yovsic;. TTJV sK9E<rr) xapovcriaas apxtKci a-rove; xapwptaK6J.tEVovc; TJ K. EYKAEIAHI:. A' 89 T.l/45

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2" 'EK9tcnt Ma91J J1UTtK<I>v---

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Mapa:yKou Kat CJt11 cruvsxsta ot J..Ul9T}'tE<; 1tapoucriacrav J..lS t11 crstpa 'tOU<; ot i(hot 'ta tpya 'tOU<;, 1toA.u cropap6t11-ra Kat U1tsu9uv6t11-ra crav �tKpoi spSUV11 'tE<;-s1ttcrn1�ovs<;.

J..lS

H SK9SO"'l ava<psp6-rav Kat CJ'tO spyo CJ1tOUOairov �a911�anKcOV 61tro<; 't01) Gauss, 't01) Pascal, 't01) Mobius, 't01) �a91l�a'ttKOU Kat xapliKTil Durer, Ka9ro<; Kat 't01) KaMt'tSXVTJ Escher 1t01) Svtro9s (j'\)xva 1tSptcrcr6-rspo KOV'tU CJ'tOU<; �a911�U'ttKOU<;. H tcr-ropia 'f(l)V Ma9Tl�a'ttKcOV sxst JlS'YUAo SVOta<pspov 6-rav ava.A.ust n<; tOES<; K1>piro<; Kat 6xt �6vo ysyov6-ra Kat ptOypa<pis<;. Ot toss<; yta TllV KMU\1'11 sv6<; sm1tsoou J..LS KavovtKii i) �11 crxfl�a-ra, 11 supsO"'l a9potcr�a-rrov, 11 SUPSO"'l KUVOVtKOnl'tOOV a1tO'tSAOUcrav 'tOV KSV'tptK6 1t1>pi)va 'f(l)V spyacrtcOV 'f(l)V �a911'tc0V Kat sKavav -ra �a811�anKa sA.K1>crnKii, Ka-ravo11-ra 011A.aoi) svota<pspoV'ta. Ot 1tapoumacrst<; -rrov �a811-rrov i)-rav s�1tA.omtcr�s<; �s videos Kat 1tpopoM<; J..lS powerpoint crmv HN 1tou sixav s-rot�acrst ot �a811-rs<; J..lS TllV Ka911'YiJ-rpta -rou<;, KaA.rocropi�ov-ra<; -rou<; S1ttO"KE1t-rs<; <rTo Ta;iat an6 1'1)V at<r61JTlK1} 1'1)� Ttxvfl� 0'1'1) A.oytK'I} Tcov Ma61JJ1«TlKrov. E1ti0"11 <; 1tpopaUov-rav Kat apxsia GeoGebra a1t6 oui<pops<; OtOaK'ttKS<; 1taps�P<icrst<; t11 <; SK1tatOSU'ttKOU.

H sK9s0"11 i)tav tva 1tap6.osty�a ota9s�anK6t11-ra<; 1tou pps9'11 Ks -ro a1t6yst6 t11 <; cr-ro "1tav-rps�a" -rrov �a811�anKrov �s t11 �oucrtKi) 6-rav -rpst<; �a9i)-rpts<; t11 <; A' -ra�11<; �s ap�6vtO, 11MK-rptKi) Kt9apa Kat -rpayouot, a1tsorocrav s�J..LS-rpa TllV spyama 1tou sixav avaA.aPst J..LS 9s�a "H �oumKi) Kat -ra KA<icr�a-ra".

H KaMt-rsxvtKT) aicr9110"11 t11 <; K MapayKou os911KS <i'l'oya �s TllV S1ttCJt1l�OvtKi) yYc00"11 1tOU a1tat-rouv -ra ota<popa �a91l�anK6. 9s�a-ra Kat -ro a1to-rsA.scr�a i)-rav 11 011�toupyia �ta<; sK9s0"11 <; 1t01) s�ota�s J..lS ayKaA.ta XPOO�U'tOOV, CJXlWU't(l)V Kat Ka'taCJKSUcOV Kat a1tS1tVSS t11 �SCJ'tacrt<i Kat TllV suxapicrt11 0"11 1tOU vtro9st Kavsi<; 6-rav ppiO"Ks-rat crs tvav 1tatxvto6-ro1to. l:s 1tsicr�a -rrov Katprov, -ra 1tatOt6. xropi<; va �00S'I'01>V 'tOU<; yovsi<; 'tOU<; SfYYUCJt11 Kav J..lS SU'tSATJ UAtKU 61tro<; xap-r6vta, K6v-rpa-1tA.aKs, 1tA.acrnKa, �uA.a, XPW�a-ra Kat MA.a, �a 1tavro a1t6 6A.a tpaA.av t11 <pavTa<Jia -rou<;, 'tO J1tpclKl 'tOU<;, 1'1J (f)lA01'lJ1ta 'tOU<; Kat t11V t1tlJ10VTa 'tOU<; yta J1tl..t1'T) a<poU 61tro<; 't6Vtcrs 11 Ka911'YfJ'tpta -rou<; «6-rav avr6 1rov 8(/Jepvav Xfleza1;6rav 6z6pfJwarJ ro brazpvav niuw 'ava Kaz 'ava, xwpi' va napanovefJovv ovre mzyp�, apKei ro tpyo rov, va eivaz tva an6

EYKAEIAm A' 89 -r.l/46

m

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2 q 'EK0£cnt MuOltpu'TtKmv-----------·

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Tr/� x.povuip> .

.t\Mcocr-rt 11 Kc:d)11rfl-rpta. -rou<; tixt 1toAA.E<; <pops<; -rovicrtt 1tro<; Ta Ma011J1«T1Ka �tv XPtla�ovTal <pnaai�1a Ka1 1tro� l>1tapxol>v at al>Ta aJ.;I] Otw�, JlOVo nol> tiva1 Kal.a Kpl>J1J1Svt� Kat 1tou yta va n<; avaKw,:(Hiftt Kavti<; XPtui�t'tat va JltA.t'tfJcrtt KaA.a -ro 9€Jla -rou.

E>tpJ.lU crurxap11'tiJpta CJ'f11V K MapayKOU rtropyia Kat CJ'tOU<; Jla911'tE<; 'f11 <;, aMa. Kat CJ'tOV �ttU9uv'tfJ 'tOU 'YUJ.lVacriou 't'll <; LKUAa<; npro1tOU K. T CJIPI'YIDTll XapaAa J17tO 1tOU CJ'f11 Pl�tt 1tUV'tO'tt 'tS'tOta tK1tatoeunKa trXttpf]Jla'ta. MaKapt 11 SK9t<rrJ 'tOU 1 ou ruJ.lVacriou LKUAa<; npro1tOU va �pti 1tOMou<; JltJ.l11'tE<;. Euxx6Jlacr-rt 11 acr-rtipeu'f11 <pav-racrA.ta Kat 11 Ot6.9t<JT) yta 1tpocr<popa 't'll <; K JlapayKOU va Jl11V t�aV'tA119ti crt au'tfJv 'f11V SK9t<JT) Kat ea 1ttptJlSvOUJ.lt 'tOU XPOVOU 'f11V 3l'l SK9t<JT) Jla&r)JlanKrov.

Ill

H SK9t<JT) 1tOU tA.a�t xropa 1tSpuqt Jlt -rhA.o: « Jlt 'tO '1'11<f>l, Jlt ip ptyA.i, � 'tO Jlapyapt-rapt. . » Kat 11 <ptnvi] tixav CJ'tOXO va Oti�ouv 01tffi<; Jla<; tt1tt 11 tK1tat0et>'ttKO<; «'ta a7to-rtMcrJ.la-ra Jlta<; ava1tro�11<; J1t96orov OtOacrKaA.ia<; Jlt �a<TIJ all11M:n•�pamlKa tK0EJ1«T«» . .

.t\Mcocr-rt Ta MaO..panKa �EV Eivcn JlOVO aaK1]0'tii; Atv Eiva1 oi>Tt Kav 11 pcnrriva TOO l>7tOMylO'p.oi> tvo� aplOJ10v. Avn0t1'� 1'« Ma�pan;Ka t;11yoiJ� � Jltyal.iJ'l'tp� a11]0� 1'00 Koapoo •

I'.TtlJill T CJIK07tOVW'l> :ExoA.tKi] l:l>Jl�ouA.o<; Ma911JlanKrov Ava-roA.tKi]<; A-rnKi]<;

EYKAEIAID; A' 89 'T.l/47

== ==

MAl.

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I:m'>po� n�ropyio1>

MA3. 0 1ttpif:pyo� TEUO'Tft � q,avTacrTdts tvav vso Jl<l6'r1JlanK6 TeA.ecrTJ1 nou O'l>Jl�oA.i�sTat JlS Kat XPllO'tJlOnotsiTat 6nro<; <paivsTat crn<; napaKaTro tO'O'tllTe<;

• • •

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#

1 # 1 =2 3 # 5 = 34 6 # 9 = 1 17 10 # 14 = 296

MtKpa KOJ.lJ.lana rupwu tonoeetouvrm mivro cre Kaee tva a1t6 ta orooeKa vo'6J.tEpa O"t11V E1ttcpciveta ev6<; KUdtKo6 poA.oywu. 'Eva� 1tovnK6<; tproet to rupi 1t01> civat t01t0eS't11 J.t£v0 1tclV(J) O'tO VO'UJ.lEpO 1 Kat Ktvo'6J.tEVo<; crUJ.lcprova J.lE Tl1 cpopa trov oetKtrov tou poA.oytou, 1tEp1tatciet tproyovta<; to rupi cre tva 1tapa tva vo'6J.lepo (tva tproet, tva acpiJvet, tva tproet, K.A.1t.). Iloto VOUJ.lEPO ea Eivat Kclt(J) a1t6 to teA.eutaio acpayroto KOJ.lJ.lan ruptou; MA2.' H Xp1>oot>A.a Kat TO Tptvo

15 19 Kat e�llYll O'Te 'tllV

BpsiTs TllV TtJlTJ Tou # anav'tll criJ crac;.

MA4. Ta 1taA.ul PtPA.ia

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peprt, rO'l'e 11 bwcpopa rwv rerpaychvmv rrov a­ plfJpdJv rmv bvo avrchv pepdJv, 1ao6rw pe ro 64-nlaaw rrtr:; bwcpopar:; rovr:;» .

H Xpooo6Aa eD.a va 1tcipEt tO tpOO 1tOI) cproya � 10:45 1t. � . Kat 1tj)bra va 1tciet J.1B tO 1tOOftA.at6 trJ<; O'tO OT<l9J.l6 tOO tp€vou, 1tOU eivm O'E Wt6cmxcrrt 6 Km J.UlKjJUi wt6 to cmitt trt<;. Ta 1tp(Ota 2 Km eivm cre avJ'l<jl6pa., ta enOJ.IBVCl 2 O'E f:1ri1tf.Oo OpOJ.IO Km ta teA.emaia 2 Km eivm O'E Katll<p<)pa. H Xpooo61.a � va OOJnrflO'et to 1t00ft­ A.at6 trJ<; J.1B tax(mrta 2 Km11 <mJV <M)cpOpa, 3 Km11 cre f:1ri1tf.Oo Km 4 1<m11 cre K<mlq>Opa. v1t06ttet 1tOl<; ea exa J.lf.ort rax6trJta 3 Km11, yt' am6 O'Korn::6et va <p6yet wt6 to cmin trJ<; � 8:40 1t. � rrtcm:Uet 6tt ea trJ<; 1tEp10'-crt\j!ow Kat 5 N::reta. va KA.etOOxret to 1tOOi)Aat6 trJ<; Km va emf3tj3a­ crtci O'tO tpt\fo. ea 1tp0Aaf3et TJ Xpooo6Aa tO tpt\fo; Nat tl 0xt Kat '}'IUri;

Iloto<; sivat o apt8JlO<; T(J)V �l�At(J)V TOU K<l81l"ffl TJl; MA5. To O'KlllJitJ.lO Tll� Tpt>1ta�

Av yw va O'Kacptet J.lta tpfuta 3m J.lTJKO<;, 3m nMtO<; Kat 3m �aeo<; wt6 2 epyate<;, XPetelSovtat 2 rope<;, 1t00'&<; rope<; XPetelSOvtat Ot· 2 c:rotoi epyate<; yta va O'Kcl­ 'I'OUV J.lta tp61ta 6m J.lTJKO<;, 6m nMtO<; Kat 6iri �eo<;, eeroprovta<; 6tt OouAeUouV J.lE tOV tOto pueJ.tO;

EYKAEIAHl: A' 89 T.l/48

------- Ta Ma&qpa'TtKO: p� AtaCJK£00:�ouv

MAl. Ta op9oyrovta Ynapxouv 3ux<popsnK6. opeoyrovta! Ilpay­ J.lan, tva� si>KoA.o� Tp6no� yta va Ta J.lETpi]­ crouJ.lE, sivat va apteJ.li]crouJ.lE T� nsptoxt� Tou napanavro crxflJ.laTo�, 6nro� <paivsTat crnt cruv€­ xsta:

37

1

2

.--+----.--+--, 3

4

5

6

7

8

10

11

9

8SKtVUJ.lE J.lE 'tO

10 opeoyrovta (2, 3, 4, 5, 6, 7, 8, 9, 10, 1 1).

Ms 2 nsptoxt� txouJ.lE 1 1 opeoyrovta (1-3, 3-4, 4-5, 3-6, 6-7, 4-7, 5-8, 7-8, 7-10, 8-1 1, 10-1 1).

M s 3 nspwxt� txouJlE 6 opeoyrovta (2-4-5, 58-9, 3-4-5, 6-7-8, 4-7-10, 5-8-1 1).

Ms 4 nspwxt� txouJ.ls 3 opeoyrovta (3-4-6-7, 4-5-7-8, 7-8-1 1-12).

Ms 5 nspwxt� txouJ.ls 3 opeoyrovta (1-2-3-4-5, 2-4-5-7-8, 4-5-7-8-9).

Ms 6 nsptoxt� txouJ.lE 2 opeoyrovta (3-4-5-67-8, 4-5-7-8-10-1 1).

Ms 7 nspwxt� txouJlE 2 opeoyrovta (2-4-5-78-10-1 1' 3-4-5-6-7 -8-9).

MA2. 0 E1tT«'JI1Jqnoc; apt9J16<;

'Ecr'tro a � y 3 y � a EtVat 0 STJ'tOUJ.lEVO� E1t'ta­ 'lfll <pt0� apteJ.l6�. A6yro Tit� EXOUJ.!E: a � y 3 y � � Kat apa A.Qyro cr'UJ.lJ.lS'tpta� ytVS'tat: � � y3y� A6yro Tit� txouJ.ls: � � y y-� y � Ot 3uva't€� 'ttJlE� 'tOU �. rocr'tS 'tO � va si­ Vat J.ltKp6Tspo i] icro TOU sivat A.pa o E1t't<l\1fi)<pw� sivat: i. y y- 1 y , ii. y 2 , iii. y A6yro Tit� Tropa TJ i Kat TJ iii anoppinToVTat, EVOO an6 Tit ii 1tpOKU1t'tEt O'tt 01tO'tS 0 sTJ­ 'tOUJ.lEVO� apteJ.l6� sivat o MA3. To �t)TOUJlEVO 'Jit)<pio Ilapantpo'6J.ls apxtKii 6n

3�.

(2)

3�.

(4)

9,

3 1 13 6 2 y y-2 6 9 3 y y-3 3 9 ( 5)

B+A=A

Kat

2-1=1 4-2=3 6-3=3 8-4=4 Kat 10-5=5 a1t6 'tOU� onoiou� J.lOVO o TEAEUTaio� txst TO B=O. A.pa A=5. Tropa an6 ntv 3s'6TEPTJ npli�TJ npocinTst 6n f=4 (50-45=5) on6Ts an6 ntV npffint npa­ �TJ EXOUJ.lE (50+45=95) Kat E1tOJ.lEvOO� A=9.

MA4. 0 otatpi:Tt)<; 8oKtJ.lUS0l>J.lE 6A.a Ta 'I'TJ<pia an6 TO 1 tro� TO 9 Kat npocrnaeouJ.lE va 30UJ.lE av J.l1tOpoUJ.lE va �pOUJ.lE 'tOU� l>1tOAot1tOU� aK€patOU�.

Ms l nspwxfl txouJ.lE

3

3 3 1, 2, 3.

-------

2= L\oKtJ.lUsOUJ.lE 'tO

7

1·3·6 9 1 .4.6 8

To Tropa 3ev J.lnopsi va sivat ytaTi sivat npro­ TO� Kat ea E1tpE1tE KU1tot0� an6 'tOU� r va sivat naA.t nou 3ev E1tt'tpE1tE'tat i) 1tOMa1tAU­ crt6 'tOU, 1tOU 3EV ytVE'tat.

8oKtJ.lUsOUJ.lE 'tO

A, B,

7

6.

3= 5

1 ·2·9 6

To Tropa 3ev J.l1topsi va sivat, yta Tov i3w A.Oyo J.lE 'tO L\oKtJ.lUsOUJ.lE 'tO

7.

6= L\oKtJ.lUsOUJ.lE TO

3

3.

1·3·8 4

4.

To nap6n sivat nproTo� txst J.lOVo'l'iJ<J>tO noUanA.acrto Kat yivs'tat:

y=9 6297926.

H npffint <JXE<JTJ sn<XAT)ewerat J.!Ovo yta B=O, evro TJ 3W'tspTJ f:xa T� f.91� &uva't� cruv3oocrJ.l�:

8.

3=

3

B-A=A.

9.

1

6=

1 .2.9 3

9=

1 ·3·6 2

T0 'tEAo� 3ev yivs'tat yta'ti ea 1tpE1tEt 'tO ytv6JlEVO J.lOVOo/fJ<ptrov va sivat 1tUAt J.lOVO'I'iJ<J>tO�

3

EYKAEIAm A' 89 T.l/49

Ta Ma9lJflllTtKci flll� AtllGKEOci�ovv

-------

1tOU 8£v yiv£Tat. Apu ot 8uvuT£c; Tt)lEc; mu fl.

fl.tuypa<pOU)l£ TU U1tOAOl1tU T£Tpaycovu O'T11V i-

civat: 2, 3 , 4, 6, 8 Kat 9

8ta ypuJ.lJ.lTJ , crTTJA11 , Ku8roc; Kat Tu y£trovtKa T£­ Tpaycovu.

Mi\5. Ta 6 a(rttpui E7t£t8i) crT11 7t£ptozi) )l£ Tu Mo T£Tpaycovu 7tp£7t£t vu T07to8£T118£i £vu um£pt, 8tuypa<pODJ.l£ TU U1tOAOt1tU T11c; CiTTJA11c;.

--1-

�T11V 1t£pto;(TJ £1tUVCD uptCiT£pa U1tUp;(£t J.llU )10VO 8£cr11 ytu VU T01t08£TTJCiOU)1£ TO U<JTEpt

'ETm O'T11V U1t6 KUTCD 7t£ptozi) U7tUp;(£t )ltU J.lOVO 8£cr11 ytu vu T07to8£Tijcrou)l£ TO ucrT£pt

J_ J

fl.tayp<i<pOU)..l£ TU UJt6A0mu T£Tpaycovu <JTllV ilita ypU)..lJ.lTJ, mTJA11, KU8ffic; Kat TU Y£lTOvtKa T£Tpaycovu.

fl.tuypa<pOU)l£ TU U1tOAOt1tU T£Tpaycovu CiTllV i­ DtU ypUJ.lJ.lTJ, crTTJA11, Ku8ffic; Kat Tu y£tTovtKa T£­ Tpaycovu.

�T11V 7t£pwxi) KaTco uptcrT£pa u7tapzct )liu J.l6vo 8£cr11 yta VU T01t08£TTJ<JOU)1£ TO UCiTEpt � T11 UlKpij 7t£PlOXTJ £7tavco U7tUPX£t )ltU J.lOVO a.

TOiT08£Tijcrou)l£ TO ucrT£pt

fl.taypa<pOU)l£ TU U1tOAOl1tU T£Tpaycovu <JT11V ifl.taypa<pOU)l£ TU U1tOAOt1tU T£Tpaycovu O'T11V i­ DtU YPUJ.lJ.lTJ , crri)A11, KU8roc; Kat m Y£lTOVtKa T£­

8ta ypuJ.lJ.lTJ, crTiJ"-11, Ku8roc; Kat Tu y£trovtKa T£­ Tpaycovu.

Tpaycovu.

T£J..o c; T07t08£TOD)l£ Kat TO T£A£muio ucrT£pt Kat £xou)l£ T11V J..ucr11 .

EYKAEIAH�

A' 89 T.l/50