2014亚洲太平洋数学奥林匹克试题与解答(英文版pdf)

XXVI Asian Paci?c Mathematics Olympiad Time allowed: 4 hours Each problem if worth 7 points Problem 1. For a positive integer m denote by S (m) and P (m) the sum and product, respectively,

of the digits of m. Show that for each positive integer n, there exist positive integers a1 , a2 , . . . , an satisfying the following conditions: S (a1 ) < S (a2 ) < · · · < S (an ) and S (ai ) = P (ai+1 ) (i = 1, 2, . . . , n). (We let an+1 = a1 .) (Proposed by the Problem Committee of the Japan Mathematical Olympiad Foundation) Problem 2. Let S = {1, 2, . . . , 2014}. For each non-empty subset T ? S , one of its members is chosen as its representative. Find the number of ways to assign representatives to all non-empty subsets of S so that if a subset D ? S is a disjoint union of non-empty subsets A, B, C ? S , then the representative of D is also the representative of at least one of A, B, C. (Proposed by Warut Suksompong, Thailand) Problem 3. Find all positive integers n such that for any integer k there exists an integer a for which a3 + a ? k is divisible by n. (Proposed by Warut Suksompong, Thailand) Problem 4. Let n and b be positive integers. We say n is b-discerning if there exists a set consisting of n di?erent positive integers less than b that has no two di?erent subsets U and V such that the sum of all elements in U equals the sum of all elements in V . (a) Prove that 8 is a 100-discerning. (b) Prove that 9 is not 100–discerning. (Proposed by the Senior Problems Committee o

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