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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. 0 Eppavov1'Ji.. 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'. 1H'H AH'A'IASft"J(H K<W •"') r• "� ../ � t�ff
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!:'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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Ot ftawtffi'tE<; KATJptKoi OJ.tro<; 'tov Ka'ttly6PlJcrav crav 6.9so Kat 'tO lla'tptapxsio 'tOV Kal..e crs <J'ttlV Krovcr'tavnvo{>1tOATJ. fta va 'tOV 'ttJ.tOO p�crst 'tou s�Tllcrav va KU'IfEt 6/.. a 'ta tpya mu. 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<;):
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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
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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.
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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;)
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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
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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
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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;
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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 �)
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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
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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 .
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� 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
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------ 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-
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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-
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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
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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
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A<; OOUJ!E �ava TI)V E�lO'OOO'll 'tOU rrpopA.iJJla'tO<; 3x + 7 = 19 . Avri va TI)V J.LE'tacrxrJJlaticrOUJ!E 01t00<; KaVaJ.LE f.lE 'tOV 1tPOllYOUJ.1EVO tp6xo, ypa<pOUJ!E O''tO rreoio EtcrayroyiJ<; 'tOU "Geogebra" OtaooxtKa -rom; n5teov<; -rwv tTVvap-rljaewv - evOeuiJv: r Tporro<;:
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Aivnaa To 1t«p«ll11A.OrP«J1JlO ABrA. I. H P«G11 TOO tivaa 10 dm K«a TO 1>'11� Too dvaa 60 cm. N« t>1tol..oy{crnt 1'0 tJ1P«Mv 1'00. 11. Av Jlf:IIDGOt>Jlt TO 1>\jlo� K«Ta 10% 1tocro o/o 1tpbtta va av;'l]crot>Jlt 1'11 P«011 yaa va 1tapa f1tivta wao To tJ1P«oov ; 5. To xcopa<pa TOO Kt>pioo n<l>pyoo f:xta Glll Jl« op9oycovioo 1t«P«ll11AorPUJ1J10t> J1t 1tU'l>p� 40 m K«l 60 m. Av 9tl.a V« pwxa omM:ma 1tOG61'111'« V1'0f1CI'm; a1tO 1'« Jl«pOOM.a, M� V1'0f.W1'� K«l Jl« pool..w 9a pcU.ta; H VTOJ1U1'« x.p�aa 35 dm1 Kaa 1 1'0 Jl«poou x.pt�aa 50 dm • 6.
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EYKAEIAID; A' 89 -r.l/2�
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I.
II.
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«.
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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
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EYKAEIAHI: A' 89 T.l/23
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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
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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]. •
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EYKAEIAm A ' 89 ,;.1/24
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llAPArONTODOI:tn;H K«I TAYTOTHTEI:
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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
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TAYTOTHTEI:
....;__ _ _ _ _ _ _
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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
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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)=
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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]
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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
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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
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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,
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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.
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3 o KptTTj ptO lGOTTtT<X� ( n - n -n )
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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
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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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llo'T£ oi>o TptyiDva dvat. toa
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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)
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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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2'1 aGKlfGlf
DO'f£ o\)o Tpiyrova dva1 ioa
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"
.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')�
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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\. EYKAEIAIIl: A ' 89·-r.l/30
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Dent o'6o Tpiyrova &ivat ioa
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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.
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·
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
-
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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
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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:
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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 •
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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.
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4,
+ + + +e +e e
e
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e � � �e� m
+ +
+
+
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+e
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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(
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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.
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n
n
(;) = n (n - I�(n - 2) .
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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
...-
-
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( 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,
---
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+
>
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.
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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
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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
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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<;.
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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
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2 q 'EK0£cnt MuOltpu'TtKmv-----------·
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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.
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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«» . .
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EYKAEIAID; A' 89 'T.l/47
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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
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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
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fl.tuypa<pOU)l£ TU U1tOAOl1tU T£Tpaycovu O'T11V i-
civat: 2, 3 , 4, 6, 8 Kat 9
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8ta ypuJ.lJ.lTJ, crTiJ"-11, Ku8roc; Kat Tu y£trovtKa T£ Tpaycovu.
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EYKAEIAH�
A' 89 T.l/50