BALKAN 1993 Problem 4

Let $p$ be a prime and $m \geq 2$ be an integer. Prove that the equation $$\frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m$$ has a positive integer solution $(x, y) \neq (1, 1)$ if and only if $m = p$ .

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