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exist k integers

Source: 2012 China TST,Test 3,Problem 2

March 25, 2012
inductioninequalitiesinequalities proposed

Problem Statement

Given an integer k≥2k\ge 2. Prove that there exist kk pairwise distinct positive integers a1,a2,…,aka_1,a_2,\ldots,a_k such that for any non-negative integers b1,b2,…,bk,c1,c2,…,ckb_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k satisfying a1≤bi≤2ai,i=1,2,…,ka_1\le b_i\le 2a_i, i=1,2,\ldots,k and ∏i=1kbici<∏i=1kbi\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i, we have k∏i=1kbici<∏i=1kbi.k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.
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