MathNet 2020 — Math Olympiad Vault — Part 01 Question 53

Given an integer $k \ge 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)! + f(x_2)! + \dots + f(x_k)!$ is divisible by $x_1! + x_2! + \dots + x_k!$ for all positive integers $x_1, x_2, \dots, x_k$.
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