Junior problems J175. Let a, b ∈ (0, π 2 ) such that sin 2 a + cos 2b ≥ 1 2 sec a and sin 2 b + cos 2a ≥ 1 2 sec b. Prove that cos 6 a + cos 6 b ≥ 1 2 . Proposed by Titu Andreescu, University of Texas at Dallas, USA J176. Solve in positive real numbers the system of equations x 1 + x 2 + ··· + x n = 1 1 x 1 + 1 x 2 + ··· + 1 x n + 1 x 1 x 2 ···x n = n 3 + 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 x + y 2 + y + z 2 + z + x 2 + 4 √ xyz ≤ 1 4 (abc + 5a + 5b + 5c). Proposed by Titu Andreescu, University of Texas at Dallas, USA J178. Find the sequences of integers (a n ) n≥0 and (b n ) n≥0 such that (2 + √ 5) n = a n + b n 1 + √ 5 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 + x 3 ) (y + z)(z 3 − x 3 ) = 3(x − y)(x 3 + y 3 ) (z + x)(x 3 − 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 3 √ a − b + 1 3 √ b − c + 1 3 √ c − d + 1 3 √ d − a = 0. 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 Mathematical Reflections 6 (2010) 1 Senior problems S175. Let p be a prime. Find all integers a 1 , . . . , a n such that a 1 + ···+ a n = p 2 −p and all solutions to the equation px n + a 1 x n−1 + ··· + a n = 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 AA 1 , BB 1 , CC 1 be cevians intersecting at P . Denote by K a = K AB 1 C 1 , K b = K BC 1 A 1 , K c = K CA 1 B 1 . Prove that K A 1 B 1 C 1 is a root of the equation x 3 + (K a + K b + K c )x 2 − 4K a K b K c = 0. Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA S177. Prove that in any acute triangle ABC, sin A 2 + sin B 2 + sin C 2 ≥ 5R + 2r 4R . Proposed by Titu Andreescu, University of Texas at Dallas, USA S178. Prove that there are sequences (x k ) k≥1 and (y k ) k≥1 of positive rational numbers such that for all positive integers n and k, (x k + y k √ 5) n = F kn−1 + F kn 1 + √ 5 2 , where (F m ) m≥1 is the Fibonacci sequence. Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania S179. Find all positive integers a and b for which (a 2 +1) 2 ab−1 is a positive integer. Proposed by Valcho Milchev, Petko Rachov Slaveikov Secondary School, Bulgaria S180. Solve in nonzero real numbers the system of equations x 4 − y 4 = 121x−122y 4xy x 4 + 14x 2 y 2 + y 4 = 122x+121y x 2 +y 2 . Proposed by Titu Andreescu, University of Texas at Dallas, USA Mathematical Reflections 6 (2010) 2 Undergraduate problems U175. What is the maximum number of points of intersection that can appear after drawing in a plane l lines, c circles, and e ellipses? Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania U176. In the space, consider the set of points (a, b, c) where a, b, c ∈ {0, 1, 2}. Find the maximum number of non-collinear points contained in the set. Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA U177. Let a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . , b n be integers greater than 1. Prove that there are infinitely many primes p such that p divides b p−1 a i i − 1 for all i = 1, 2, . . . , n. Proposed by Gabriel Dospinescu, Ecole Normale Superieure, France U178. Let k be a fixed positive integer and let S (j) n = n j + n j+k + n j+2k + ··· , j = 0, 1, . . . , k − 1. Prove that S (0) n + S (1) n cos 2π k + ··· + S (k−1) n cos 2(k −1)π k 2 + S (1) n sin 2π k + S (2) n sin 4π k + ··· + S (k−1) n sin 2(k −1)π k 2 = 2 cos π k 2n . Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania U179. Let f : [0, ∞] → R be a continuous function such that f(0) = 0 and f (2x) ≤ f(x) + x for all x ≥ 0. Prove that f(x) < x for all x ∈ [0, ∞]. Proposed by Samin Riasat, University of Dhaka, Bangladesh U180. Let a 1 , . . . , a k , b 1 , . . . , b k , n 1 , . . . , n k be positive real numbers and a = a 1 + ··· + a k , b = b 1 + ··· + b k , n = n 1 + ··· + n k , k ≥ 2. Prove that 1 0 (a 1 + b 1 x) n 1 ···(a k + b k x) n k dx ≤ (a + b) n+1 − a n+1 (n + 1)b . Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania Mathematical Reflections 6 (2010) 3 Olympiad problems O175. Find all pairs (x, y) of positive integers such that x 3 − y 3 = 2010(x 2 + y 2 ). Proposed by Titu Andreescu, University of Texas at Dallas, USA O176. Let P (n) be the following statement: for all positive real numbers x 1 , x 2 , . . . , x n such that x 1 + x 2 + ··· + x n = n, x 2 √ x 1 + 2x 3 + x 3 √ x 2 + 2x 4 + ··· + x 1 √ x n + 2x 2 ≥ n √ 3 . Prove that P (n) is true for n ≤ 4 and false for n ≥ 9. Proposed by Gabriel Dospinescu, Ecole Normale Superieure, France O177. Let P be point situated in the interior of a circle. Two variable perpendicular lines through P intersect the circle at A and B. Find the locus of the midpoint of the segment AB. Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania O178. Let m and n be positive integers. Prove that for each odd positive integer b there are infinitely many primes p such that p n ≡ 1 (mod b) m implies b m−1 | n. Proposed by Vahagn Aslanyan, Yerevan, Armenia O179. Prove that any convex quadrilateral can be dissected into n ≥ 6 cyclic quadrilaterals. Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania O180. Let p be a prime. Prove that each positive integer n ≥ p, p 2 divides n+p p 2 − n+2p 2p − n+p 2p . Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania Mathematical Reflections 6 (2010) 4 . 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 + x 3 ) (y + z)(z 3 − x 3 ) = 3( x − y)(x 3 + y 3 ) (z. x)(x 3 − 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 3 √ a − b + 1 3 √ b − c + 1 3 √ c. b + 1 3 √ b − c + 1 3 √ c − d + 1 3 √ d − a = 0. 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 Mathematical