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Problem 504


Problem 504


Square on the Inside

Let ABCD be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows:

A(a, 0), B(0, b), C(?c, 0), D(0, ?d), where 1 ≤ a, b, c, d ≤ m and a, b, c, d, m are integers.

It can be shown that for m = 4 there are exactly 256 valid ways to construct ABCD. Of these 256 quadrilaterals, 42 of them strictly contain a square number of lattice points.

How many quadrilaterals ABCD strictly contain a square number of lattice points for m = 100?


平方数个内部格点

四边形ABCD的四个顶点都是坐标轴上的格点,其坐标分别是:

A(a, 0), B(0, b), C(?c, 0), D(0, ?d),其中1 ≤ a, b, c, d ≤ m,且a, b, c, d, m均为整数。

可以验证,对于m = 4,有256种构造ABCD的方式,在这256个四边形中,有42个严格地包含了平方数个格点在其内部。

对于m = 100,有多少个四边形ABCD严格地包含了平方数个格点在其内部?