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Çàêëþ÷èòåëüíûé ýòàï XXXVI Âñåðîññèéñêîé îëèìïèàäû øêîëüíèêîâ ïî ìàòåìàòèêå ÎËÈÌÏÈÀÄÛ Çàêëþ÷èòåëüíûé ýòàï Âñåðîññèéñêîé ìàòåìàòè÷åñêîé îëèì- ïèàäû ïðîøåë ñ 25 ïî 30 àïðåëÿ â ãîðîäå Ìàéêîïå. Çà ïîñëåäíèå 15 ëåò ñòîëèöà Ðåñïóáëèêè Àäûãåÿ óæå â ÷åòâåð- òûé ðàç ïðèíèìàåò ãëàâíîå ìåðîïðèÿòèå ãîäà äëÿ ñàìûõ òàëàíòëèâûõ þíûõ ìàòåìàòèêîâ ñî âñåé Ðîññèè. Îëèìïèàäû â Ìàéêîïå âñåãäà ïðåêðàñíî îðãàíèçîâàíû, ïðîõîäÿò â äðóæåñêîé àòìîñôåðå ñîâìåñòíîé ðàáîòû îðãêîìèòåòà, æþðè, ðóêîâîäèòåëåé êîìàíä è ó÷àñòíèêîâ. Îñîáî áëàãîäàðèì çà ìíîãîëåòíþþ ðàáîòó â ìàòåìàòè÷åñêîì îáðàçîâàíèè è îðãà- íèçàöèè îëèìïèàä äåêàíà ôàêóëüòåòà ìàòåìàòèêè è êîìïü- þòåðíûõ íàóê Àäûãåéñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà Äàóäà Êàçáåêîâè÷à Ìàìèé. Èçìåíåíèÿ â ïîðÿäêå ïðîâåäåíèÿ îëèìïèàäû ïîçâîëèëè â ýòîì ãîäó ó÷àñòâîâàòü â çàêëþ÷èòåëüíîì ýòàïå ïðàêòè÷åñ- êè âñåì ñèëüíåéøèì îëèìïèéöàì ñòðàíû. Ïðè ýòîì êàæäûé ðåãèîí Ðîññèè èìåë ïðàâî âûñòàâèòü õîòÿ áû îäíîãî øêîëü- íèêà. Âûðîñëî è îáùåå êîëè÷åñòâî ó÷àñòíèêîâ îëèìïèàäû ïî ñðàâíåíèþ ñ ïðåäûäóùèìè ãîäàìè â íåé ïðèíÿëè ó÷àñòèå 90 äåâÿòèêëàññíèêîâ, 64 äåñÿòèêëàññíèêà è 98 îäèííàäöàòèêëàññíèêîâ. Ïî òðàäèöèè Âñåðîññèéñêàÿ îëèì- ïèàäà ïðèíèìàëà ãîñòåé êîìàíäû Áîëãàðèè è Êèòàÿ. Íà òîðæåñòâåííîé öåðåìîíèè çàêðûòèÿ îëèìïèàäû ïðè- ñóòñòâîâàë Ïðåçèäåíò Ðåñïóáëèêè Àäûãåÿ À.Ê.Òõàêóøèíîâ, âðó÷èâøèé íàãðàäû ëó÷øèì ó÷àñòíèêàì. Ñïåöèàëüíûõ ïðè- çîâ óäîñòîèëèñü Ìèõàèë Ãðèãîðüåâ (Êàçàíü), ßðîñëàâ Ñåð- ãèåíêî (Êðàñíîäàð) è Âàñèëèé Ìîêèí (Ñàðàòîâ), ïîêàçàâ- øèå ëó÷øèå ðåçóëüòàòû â ïàðàëëåëÿõ 9, 10 è 11 êëàññà. Ïî ðåçóëüòàòàì Âñåðîññèéñêîé îëèìïèàäû èç ÷èñëà êàí- äèäàòîâ áûëà ñôîðìèðîâàíà íàöèîíàëüíàÿ êîìàíäà Ðîñ- ñèè íà Ìåæäóíàðîäíóþ ìàòåìàòè÷åñêóþ îëèìïèàäó â ñî- ñòàâå: Âàñèëèé Ìîêèí (Ñàðàòîâ), Ìàðñåëü Ìàòäèíîâ (Îðåí- áóðã, ÑÓÍÖ ÌÃÓ), Ñòàíèñëàâ Åðîõèí (Ñàíêò-Ïåòåðáóðã), Âèêòîð Îìåëüÿíåíêî (Áåëãîðîä), Ôåäîð Èâëåâ (Òðîèöê, ÑÓÍÖ ÌÃÓ), Êèðèëë Ñàâåíêîâ (Ñàíêò-Ïåòåðáóðã). Îïðîñ îïðåäåëèë ëó÷øèå, ïî ìíåíèþ øêîëüíèêîâ, çàäà÷è îëèìïèàäû. Îí ïîêàçàë, ÷òî ó÷àñòíèêàì â ýòîì ãîäó áîëüøå ïîíðàâèëèñü òðóäíûå çàäà÷è: â ïàðàëëåëè 9 êëàññà ïðèçî- âûå ìåñòà ïîëó÷èëè çàäà÷è 7, 8 è 4, â ïàðàëëåëè 10 êëàññà 8, 7 è 3, â ïàðàëëåëè 11 êëàññà ïåðâîå ìåñòî ïîäåëèëè çàäà÷è 7, 8, à â «ïðèçåðàõ» îêàçàëèñü çàäà÷è 3 è 4. Ïóáëèêóåì óñëîâèÿ çàäà÷ è ñïèñîê ïîáåäèòåëåé è ïðèçå- ðîâ Çàêëþ÷èòåëüíîãî ýòàïà XXXVI Âñåðîññèéñêîé îëèìïèà- äû øêîëüíèêîâ ïî ìàòåìàòèêå. Çàäà÷è çàêëþ÷èòåëüíîãî ýòàïà 9 êëàññ 1. Èìååòñÿ 24 êàðàíäàøà ÷åòûðåõ öâåòîâ ïî 6 êàðàíäà- øåé êàæäîãî öâåòà. Èõ ðàçäàëè 6 ðåáÿòàì òàê, ÷òî êàæäûé ïîëó÷èë ïî 4 êàðàíäàøà. Êàêîå íàèìåíüøåå êîëè÷åñòâî ðåáÿò âñåãäà ìîæíî âûáðàòü, ÷òîáû ó íèõ ãàðàíòèðîâàííî íàøëèñü êàðàíäàøè âñåõ öâåòîâ, âíå çàâèñèìîñòè îò ðàñïðå- äåëåíèÿ êàðàíäàøåé? È.Áîãäàíîâ, Î.Ïîäëèïñêèé 2. Ïî îêðóæíîñòè ðàññòàâëåíî 100 ïîïàðíî ðàçëè÷íûõ ÷èñåë. Äîêàæèòå, ÷òî ìîæíî âûáðàòü 4 ïîäðÿä ñòîÿùèõ ÷èñëà òàêèì îáðàçîì, ÷òîáû ñóììà äâóõ êðàéíèõ ÷èñåë ýòîé ÷åòâåðêè áûëà ñòðîãî áîëüøå ñóììû ñðåäíèõ. Ñ.Áåðëîâ 3. Ïðÿìûå, êàñàþùèåñÿ îêðóæíîñòè ω â òî÷êàõ A è B, ïåðåñåêàþòñÿ â òî÷êå O. Òî÷êà I öåíòð ω . Íà ìåíüøåé äóãå AB îêðóæíîñòè ω âûáðàíà òî÷êà C, îòëè÷íàÿ îò ñåðåäèíû äóãè. Ïðÿìûå AC è OB ïåðåñåêàþòñÿ â òî÷êå D, à ïðÿìûå BC è OA â òî÷êå E. Äîêàæèòå, ÷òî öåíòðû îïèñàííûõ îêðóæíîñòåé òðåóãîëüíèêîâ ACE, BCD è OCI ëåæàò íà îäíîé ïðÿìîé. À.Ïîëÿíñêèé 4. Ñì. çàäà÷ó Ì2194 «Çàäà÷íèêà «Êâàíòà». 5. Ðàçëè÷íûå äåéñòâèòåëüíûå ÷èñëà a è b òàêîâû, ÷òî óðàâíåíèå ( )( ) 2 2 20 10 20 10 0 x ax b x bx a + + + + = íå èìååò êîðíåé. Äîêàæèòå, ÷òî ÷èñëî ( ) 20 b a íå ÿâëÿåòñÿ öåëûì. Ï.Êîçëîâ 6. Ó êàæäîãî èç 1000 ãíîìîâ åñòü êîëïàê, ñèíèé ñíàðóæè è êðàñíûé âíóòðè (èëè íàîáîðîò). Åñëè íà ãíîìå íàäåò êðàñíûé êîëïàê, òî îí ìîæåò òîëüêî ëãàòü, à åñëè ñèíèé òîëüêî ãîâîðèòü ïðàâäó. Íà ïðîòÿæåíèè îäíîãî äíÿ êàæäûé ãíîì ñêàçàë êàæäîìó «Íà òåáå êðàñíûé êîëïàê!» (ïðè ýòîì íåêîòîðûå ãíîìû â òå÷åíèå äíÿ âûâîðà÷èâàëè ñâîé êîëïàê íàèçíàíêó). Íàéäèòå íàèìåíüøåå âîçìîæíîå êîëè÷åñòâî âûâîðà÷èâàíèé. È.Áîãäàíîâ 7. Íàçîâåì íàòóðàëüíîå ÷èñëî n íåóäà÷íûì, åñëè åãî íåëüçÿ ïðåäñòàâèòü â âèäå 2 2 1 1 x n y = ïðè íàòóðàëüíûõ xy > 1. Êîíå÷íî èëè áåñêîíå÷íî êîëè÷åñòâî íåóäà÷íûõ ÷èñåë? Â.Ñåíäåðîâ 8.  îñòðîóãîëüíîì òðåóãîëüíèêå ABC ìåäèàíà AM äëèí- íåå ñòîðîíû AB. Äîêàæèòå, ÷òî òðåóãîëüíèê ABC ìîæíî ðàçðåçàòü íà òðè ÷àñòè, èç êîòîðûõ ñêëàäûâàåòñÿ ðîìá. C.Âîë÷¸íêîâ 10 êëàññ 1. Èìååòñÿ 40 êàðàíäàøåé ÷åòûðåõ öâåòîâ ïî 10 êàðàí- äàøåé êàæäîãî öâåòà. Èõ ðàçäàëè 10 ðåáÿòàì òàê, ÷òî êàæäûé ïîëó÷èë ïî 4 êàðàíäàøà. Êàêîå íàèìåíüøåå êîëè- ÷åñòâî ðåáÿò âñåãäà ìîæíî âûáðàòü, ÷òîáû ó íèõ ãàðàíòèðî- âàííî íàøëèñü êàðàíäàøè âñåõ öâåòîâ, âíå çàâèñèìîñòè îò ðàñïðåäåëåíèÿ êàðàíäàøåé? È.Áîãäàíîâ, Î.Ïîäëèïñêèé 2. Ñì. çàäà÷ó 2 äëÿ 9 êëàññà. 50-59.p65 13.10.10, 13:18 50

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Page 10: Страницы 50–59

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