mustaqil yechish uchun 9-sinf

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bahodir kamolov, ne’matjon kamalov matematikadan bilimlar bellashuvi va olimpiada masalalari 9-sinf. mustaqil yechish uchun masalalar 1. 1,n n nx y z n    tenglamani natural sonlarda yeching. 2. , , 0a b c  bo‘lgan hol uchun ushbu 1 1 1 1 1 1 a b c ab ac bc      tengsizlikni isbotlang. 3. trapetsiyaning bitta asosiga yopishgan burchaklarining yig‘indisi 900 ga teng bo‘lsa, asoslarining o‘rtalarini tutashtiruvchi kesma ular ayirmasining yarmiga teng ekanligini isbotlang. 4. teng yonli trapetsiyaning asoslari a va b , yon tomoni c va diagonali d ga teng bo‘lsa, 2d ab c  ekanini isbotlang. 5. agar ,a b  ratsional sonlar bo‘lib, a b c  tenglik o‘rinli bo‘lsa, u holda a va b lar ham ratsional bo‘lishini isbotlang. 6. , ,a b c -natural sonlar ko‘rsatilgan tartibda geometrik progressiyani tashkil qilsa,     2 ; ;ekub a …

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                (bunda a a    ning butun qisimi). 16. hisoblang: 2 2 2 2 2 2 1 1 1 1 1 1 1 1 ... 1 ? 2 3 3 4 2011 2012           17.     1 1 2 2 3 3, , , ,x y x y x y sonlari 3 23 2010 x xy  va 3 23 2009 y x y  tenglamaning ildizlari bo‘lsa, 1 2 3 1 2 3 1 1 1 x x x y y y                        ni toping. 18. uchburchakning eng katta tomoni 5 sm, eng kichigi 1 sm. bu uchburchak …

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ning bo‘luvchisi bo‘lsa, shunday d larning eng katta natural qiymati topilsin. 29. burchaklari tub sonlar bilan ifodalanuvchi nechta uchburchak mavjud? 30. ushbu  1982 1983 1 k kn n    tenglamaning barcha butun yechimlarni toping. 31. abc uchburchakning tomonlari 6 2ab  , 8bc  , 14ac  ga teng. uchburchakning b uchidan mediana va balandlik o‘tkazilgan. c uchidan o‘tkazilgan p to‘g‘ri chiziq mediananing davomini k nuqtada, balandlikning davomini m nuqtada va ba ning davomini n nuqtada kesadi. km cn nisbat eng katta bo‘lishi uchun p va ac to‘g‘ri chiziqlar orasidagi burchak nimaga teng bo‘lishi kerak? 32. abc uchburchakning ichida ixtiyoriy m nuqta olingan va bu nuqtadan , ,am bm cm to‘g‘ri chiziqlar o‘tkazilgan. bu to‘g‘ri chiziqlar uchburchak tomonlarini, mos ravishda 1 1 1, ,a b c nuqtalarda kesib o‘tadi. quyidagi tengsizlikni isbotlang: 1 1 1 6 am bm cm am b m c m    …

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lari bo‘lsa, (c gipotenuza) ushbu 35 ( ) 4 ab a b c c   tengsizlikni isbotlang. 39. [ ] [2 ] [3 ] 6x x x   tenglamani yeching, bu yerda [ ]x -x sonning butun qismi. 40. , , 0a b c  va 2 2 2 5 3 a b c   bo‘lsa, 1 1 1 1 a b c abc    tengsizlikni isbotlang. 41. agar , 0a b  va [ , ], ( 1, )ix a b i n  bo‘lsa, quyidagi tengsizlikni isbotlang: 2 1 2 1 2 1 1 1 ( ... ) ...n n b x x x n x x x a              42. taqqoslang. 1 2 3 4 5 6 ... 2015 2016        va 2017 1008 …

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bahodir kamolov, ne’matjon kamalov matematikadan bilimlar bellashuvi va olimpiada masalalari 9-sinf. mustaqil yechish uchun masalalar 1. 1,n n nx y z n    tenglamani natural sonlarda yeching. 2. , , 0a b c  bo‘lgan hol uchun ushbu 1 1 1 1 1 1 a b c ab ac bc      tengsizlikni isbotlang. 3. trapetsiyaning bitta asosiga yopishgan burchaklarining yig‘indisi 900 ga teng bo‘lsa, asoslarining o‘rtalarini tutashtiruvchi kesma ular ayirmasining yarmiga teng ekanligini isbotlang. 4. teng yonli trapetsiyaning asoslari a va b , yon tomoni c va diagonali d ga teng bo‘lsa, 2d ab c  ekanini isbotlang. 5. agar ,a b  ratsional sonlar bo‘lib, a b c  tenglik o‘rinli bo‘lsa, u …

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