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There are exactly k squares in T having these two points

Source: Balkan MO 1995, Problem 4

April 24, 2006
analytic geometryinductioncombinatorics proposedcombinatorics

Problem Statement

Let nn be a positive integer and S\mathcal S be the set of points (x,y)(x, y) with x,y∈{1,2,…,n}x, y \in \{1, 2, \ldots , n\}. Let T\mathcal T be the set of all squares with vertices in the set S\mathcal S. We denote by aka_k (k≥0k \geq 0) the number of (unordered) pairs of points for which there are exactly kk squares in T\mathcal T having these two points as vertices. Prove that a0=a2+2a3a_0 = a_2 + 2a_3.
Yugoslavia
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