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B1

Part of 2019 Putnam

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Putnam 2019 B1

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12/10/2019
Denote by Z2\mathbb Z^2 the set of all points (x,y)(x,y) in the plane with integer coordinates.  For each integer n≥0n\geq 0, let PnP_n be the subset of Z2\mathbb Z^2 consisting of the point (0,0)(0,0) together with all points (x,y)(x,y) such that x2+y2=2kx^2+y^2=2^k for some integer k≤nk\leq n.  Determine, as a function of nn, the number of four-point subsets of PnP_n whose elements are the vertices of a square.
PutnamPutnam 2019