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Permutations of (1,...,n) such that sum of a_i/i is integer

Source: Indonesian Mathematics Olympiad 2011, Day 1, Problem 2

September 13, 2011
inductionnumber theory proposed

Problem Statement

For each positive integer nn, let sns_n be the number of permutations (a1,a2,⋯ ,an)(a_1, a_2, \cdots, a_n) of (1,2,⋯ ,n)(1, 2, \cdots, n) such that a11+a22+⋯+ann\dfrac{a_1}{1} + \dfrac{a_2}{2} + \cdots + \dfrac{a_n}{n} is a positive integer. Prove that s2n≥ns_{2n} \ge n for all positive integer nn.
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