ILL 1977 Problem 15

Let $n$ be an integer greater than $1$ . In the Cartesian coordinate system we consider all squares with integer vertices $(x,y)$ such that $1\le x,y\le n$ . Denote by $p_k\ (k=0,1,2,\ldots )$ the number of pairs of points that are vertices of exactly $k$ such squares. Prove that $\sum_k(k-1)p_k=0$ .