Junior problems - Phần 3

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Junior problems - Phần 3 Junior problems J175. Let a, b ∈ (0, π ) such that sin2 a + cos 2b ≥ 1 1 sec a and sin2 b + cos 2a ≥ sec b. Prove that 2 2 2 1 cos6 a + cos6 b ≥ . 2 Proposed by Titu Andreescu, University of Texas at Dallas, USA J176. Solve in positive real numbers the system of equations x1 + x2 + · · · + xn = 1 1 1 1 1 3 x1 + x2 + · · · + xn + x1 x2 ···xn = n + 1. 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 z+x √ x+y y+z 1 + 4 xyz ≤ (abc + 5a + 5b + 5c). + + 2 2 2 4 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 √ √n 1+ 5 (2 + 5) = an + bn 2 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 )(y 3 − z 3 ) = 3(z − x)(z 3 + x3 )  (y + z )(z 3 − x3 ) = 3(x − y )(x3 + y 3 )  (z + x)(x3 − y 3 ) = 3(y − z )(y 3 + z 3 )  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 √ +√ +√ +√ = 0. 3 3 3 3 a−b b−c c−d d−a √ √ √ √ Prove that 3 a − b + 3 b − c + 3 c − d + 3 d − a = 0. Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania 1 Mathematical Reflections 6 (2010) 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 + a1 xn−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 = KAB1 C1 , Kb = KBC1 A1 , Kc = KCA1 B1 . Prove that KA1 B1 C1 is a root of the equation x3 + (Ka + Kb + Kc )x2 − 4Ka Kb Kc = 0. Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA S177. Prove that in any acute triangle ABC, A B C 5R + 2r + sin + sin ≥ sin . 2 2 2 4R Proposed by Titu Andreescu, University of Texas at Dallas, USA S178. 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+ 5 (xk + yk 5) = Fkn−1 + Fkn , 2 ...