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10 divides difference of products of powers

Source: European Mathematical Cup, 2015, Senior, P1

December 30, 2016
number theorycombinatoricspigeonhole principle

Problem Statement

A={a,b,c}A = \{a, b, c\} is a set containing three positive integers. Prove that we can find a set B⊂AB \subset A, B={x,y}B = \{x, y\} such that for all odd positive integers m,nm, n we have 10∣xmyn−xnym.10\mid x^my^n-x^ny^m.
Tomi Dimovski
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