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x_1+x_2+...+x_k is divided by k for all k with 1<=k<=n

Source: 2022 Argentina OMA Finals L3 p2

March 25, 2024
number theory

Problem Statement

Determine all positive integers nn such that numbers from 11 to nn can be sorted in some order x1,x2,...,xnx_1,x_2,...,x_n with the property that the number x1+x2+...+xkx_1+x_2+...+x_k is divisible by kk, for all 1≤k≤n1\le k\le n., that is 11 is divides x1x_1, 22 divides x1+x2x_1+x_2, 33 divides x1+x2+x3x_1+x_2+x_3, and so on until nn divides x1+x2+...+xnx_1+x_2+...+x_n.
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