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 $(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A,.., (A + n)^2- A$ is a multiple of $n$.
Source: 2018 Romania JBMO TST 5.1
Tags: number theory, Perfect Square
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 $(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A,.., (A + n)^2- A$ is a multiple of $n$.