MathDB
Two permutations

Source: Iran prepration exam

April 24, 2006
abstract algebragroup theorycombinatoricspermutationsIMO Shortlist

Problem Statement

Suppose that a1 a_1, a2 a_2, … \ldots, an a_n are integers such that n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n. Prove that there exist two permutations (b1,b2,…,bn) \left(b_1,b_2,\ldots,b_n\right) and (c1,c2,…,cn) \left(c_1,c_2,\ldots,c_n\right) of (1,2,…,n) \left(1,2,\ldots,n\right) such that for each integer i i with 1≤i≤n 1\leq i\leq n, we have n\mid a_i \minus{} b_i \minus{} c_i
Proposed by Ricky Liu & Zuming Feng, USA
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