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Problem 292


Problem 292


Pythagorean Polygons

We shall define a pythagorean polygon to be a convex polygon with the following properties:

  • there are at least three vertices,
  • no three vertices are aligned,
  • each vertex has integer coordinates,
  • each edge has integer length.

For a given integer n, define P(n) as the number of distinct pythagorean polygons for which the perimeter is ≤ n.
Pythagorean polygons should be considered distinct as long as none is a translation of another.

You are given that P(4) = 1, P(30) = 3655 and P(60) = 891045.
Find P(120).


毕达哥拉斯多边形

我们定义毕达哥拉斯多边形为满足下列性质的凸多边形

  • 有至少三个顶点,
  • 不存在三个顶点共线,
  • 每个顶点坐标均为整数
  • 每条边长度均为整数

对于给定的整数n,记P(n)为边长≤ n的不同毕达哥拉斯多边形的数目。
只要不能将其中一个通过平移得到另一个,就被认为是不同的毕达哥拉斯多边形。

已知P(4) = 1,P(30) = 3655,以及P(60) = 891045。
求P(120)。