Question 31
Determine all the natural numbers $n$ for which there are $2n$ different positive integers $x_1, \dots, x_n, y_1, \dots, y_n$ such that the product
$$
(11x_1^2 + 12y_1^2)(11x_2^2 + 12y_2^2) \dots (11x_n^2 + 12y_n^2)
$$
is a perfect square.
$$
(11x_1^2 + 12y_1^2)(11x_2^2 + 12y_2^2) \dots (11x_n^2 + 12y_n^2)
$$
is a perfect square.