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Problem 755

AIME late

Number theory Difficulty 5.4 Prove it

1. Natural numbers $a, b, c$ and $d$ are such that $2015^{a}+2015^{b}=2015^{c}+2015^{d}$ . Can the numbers $a^{2015}+b^{2015}$ and $c^{2015}+d^{2015}$ be different?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

1. See the solution to problem 4 for 8th grade.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.