KöMaL problem A, 2015

Let $n\ge2$ be an integer. Prove that there exist integers $a_1,\dots,a_{n-1}$ such that $a_1 \arctg 1 + a_2 \arctg 2 +\ldots+ a_{n-1}\arctg(n-1) = \arctg n$ if and only if $n^2+1$ divides $(1^2+1)(2^2+1)\ldots\big((n-1)^2+1\big)$.
Based on a problem of IMC 2015, Blagoevgrad
(5 pont)