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Asian Pacific Mathematical Olympiad 2010 Problem 2

Source:

May 7, 2010
number theoryAPMOPerfect PowersAdditive Number Theory

Problem Statement

For a positive integer k,k, call an integer a purepure k−thk-th powerpower if it can be represented as mkm^k for some integer m.m. Show that for every positive integer n,n, there exists nn distinct positive integers such that their sum is a pure 2009−2009-th power and their product is a pure 2010−2010-th power.
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