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1^1 + 3^3 + 5^5 + ... + (2^{2013} - 1)^{(2^{2013} - 1)} / 2^{2013} is odd

Source: 2013 Saudi Arabia BMO TST I p8

July 24, 2020
algebraSumoddnumber theory

Problem Statement

Prove that the ratio 11+33+55+...+(22013−1)(22013−1)22013\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}} is an odd integer.
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