Question 85
Prove that a positive integer $A$ is a perfect square if and only if, for all positive integers $n$, at least one of the numbers
$$
(A+1)^2 - A, (A+2)^2 - A, (A+3)^2 - A, \dots, (A+n)^2 - A
$$
is a multiple of $n$.
$$
(A+1)^2 - A, (A+2)^2 - A, (A+3)^2 - A, \dots, (A+n)^2 - A
$$
is a multiple of $n$.