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Divisibility in function

Source: Balkan MO Shortlist 2020 N3

September 14, 2021
functionnumber theoryAZE EGMO TST

Problem Statement

Given an integer k≥2k\geq 2, determine all functions ff from the positive integers into themselves such that f(x1)!+f(x2)!+⋯f(xk)!f(x_1)!+f(x_2)!+\cdots f(x_k)! is divisibe by x1!+x2!+⋯xk!x_1!+x_2!+\cdots x_k! for all positive integers x1,x2,⋯xkx_1,x_2,\cdots x_k.
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