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Tham khảo tài liệu 'junior problems - phần 3', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Junior problems J175. J176. Let a b 2 0 2 such that sin2 a cos 2b 2 sec a and sin2 b cos 2a 2 sec b. Prove that cos6 a cos6 b . Proposed by Titu Andreescu University of Texas at Dallas USA Solve in positive real numbers the system of equations X1 X2 --- Xn 1 p- P P V T n3 1. X1 X2 Xn X1X2-Xn Proposed by Neculai Stanciu George Emil Palade Secondary School Buzau Romania J177. Let x y z be nonnegative real numbers such that ax by cz 3abc for some positive real numbers a b c. Prove that x y y z z 2 x pxyz 4 abc 5a 5b 5c . Proposed by Titu Andreescu University of Texas at Dallas USA J178. Find the sequences of integers an n 0 and bn n 0 such that 2 p5 n an bn1 2 5 for each n 0. Proposed by Dorin Andrica Babes-Bolyai University Cluj-Napoca Romania J179. Solve in real numbers the system of equations x y y3 z3 3 z x z3 x3 y z z3 x3 3 x y x3 y3 z x x3 - y3 3 y - z y3 z3 Proposed by Titu Andreescu University of Texas at Dallas USA J180. Let a b c d be distinct real numbers such that 1 1 1 1 p a b p b c p c d p d a Prove that pa b pb c pc d pd a 0. Proposed by Dorin Andrica Babes-Bolyai University Cluj-Napoca Romania Mathematical Reflections 6 2010 1 Senior problems S175. Let p be a prime. Find all integers a1 . an such that a1 ------- an p2 p and all solutions to the equation pxn a1xn 1 an 0 are nonzero integers. Proposed by Titu Andreescu University of Texas at Dallas USA and Dorin Andrica Babes-Bolyai University Cluj-Napoca Romania S176. Let ABC be a triangle and let AA1 BB1 CC1 be cevians intersecting at P. Denote by Ka Kab1C1 Kb KBC1A1 Kc Kca1B1 . Prove that Ka1B1C1 is a root of the equation x3 Ka Kb Kc x2 - 4KaKbKc 0. Proposed by Ivan Borsenco Massachusetts Institute of Technology USA 5177. Prove that in any acute triangle ABC sin A sin B sin C 2 2 2 4R Proposed by Titu Andreescu University of Texas at Dallas USA 5178. Prove that there are sequences xk k 1 and yk k 1 of positive rational numbers such that for all positive integers n and k n 1 75 xk y 5 Fkn-1 Fkn------2
