Problem

Source: Stars of Mathematics 2019, Senior, P2

Tags: number theory



If $n\geqslant 3$ is an integer and $a_1,a_2,\dotsc ,a_n$ are non-zero integers such that $$a_1a_2\cdots a_n\left( \frac{1}{a_1^2}+\frac{1}{a_2^2} +\cdots +\frac{1}{a_n^2}\right)$$is an integer, does it follow that the product $a_1a_2\cdots a_n$ is divisible by each $a_i^2$?