Question 173
Let $a_1, a_2, \dots, a_n$ be non-zero integers such that
$$
a_1 a_2 \dots a_n \left( \frac{1}{a_1^2} + \frac{1}{a_2^2} + \dots + \frac{1}{a_n^2} \right)
$$
is an integer. Prove that $a_k^2 \mid a_1 a_2 \dots a_n$ for all $k = 1, 2, \dots, n$.
$$
a_1 a_2 \dots a_n \left( \frac{1}{a_1^2} + \frac{1}{a_2^2} + \dots + \frac{1}{a_n^2} \right)
$$
is an integer. Prove that $a_k^2 \mid a_1 a_2 \dots a_n$ for all $k = 1, 2, \dots, n$.