Problem 397
Triangle on parabola
On the parabola y = x2/k, three points A(a, a2/k), B(b, b2/k) and C(c, c2/k) are chosen.
Let F(K, X) be the number of the integer quadruplets (k, a, b, c) such that at least one angle of the triangle ABC is 45-degree, with 1 ≤ k ≤ K and -X ≤ a < b < c ≤ X.
For example, F(1, 10) = 41 and F(10, 100) = 12492.
Find F(106, 109).
双曲线上的三角形
在双曲线上y = x2/k上选择三个点A(a, a2/k)、B(b, b2/k)和C(c, c2/k)。
若四元整数组(k, a, b, c)满足1 ≤ k ≤ K以及-X ≤ a < b < c ≤ X，且使得三角形ABC至少有一个角为45度，记这样的四元整数组的总数为F(K, X)。
已知F(1, 10) = 41以及F(10, 100) = 12492。
求F(106, 109)。