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equal sum of the $p$-th powers of numbers from 1,2,...,2^n

Source: 1996 Swedish Mathematical Competition p5

April 2, 2021
number theorypower of 2Sum of powers

Problem Statement

Let n≥1n \ge 1. Prove that it is possible to select some of the integers 1,2,...,2n1,2,...,2^n so that for each p=0,1,...,n−1p = 0,1,...,n - 1 the sum of the pp-th powers of the selected numbers is equal to the sum of the pp-th powers of the remaining numbers.
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