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2017 Iran MO (3rd round), 1

Let $x$ and $y$ be integers and let $p$ be a prime number. Suppose that there exist realatively prime positive integers $m$ and $n$ such that $$x^m \equiv y^n \pmod p$$ Prove that there exists an unique integer $z$ modulo $p$ such that $$x \equiv z^n \pmod p \quad \text{and} \quad y \equiv z^m \pmod p$$

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