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1^1, 3^3, 5^5, ..., (2n-1)^{2n-1} different remainders divided by 2^n

Source: Switzerland - 2022 Swiss Final Round p2

November 17, 2022
number theoryremainderdivides

Problem Statement

Let nn be a positive integer. Prove that the numbers 11,33,55,...,(2n−1)2n−11^1, 3^3, 5^5, ..., (2n-1)^{2n-1} all give different remainders when divided by 2n2^n.
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