Triangle on parabola

Published on Sunday, 7th October 2012, 05:00 am; Solved by 229;
Difficulty rating: 70%

Problem 397

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 ≤ kK and -Xa < b < cX.

For example, F(1, 10) = 41 and F(10, 100) = 12492.
Find F(106, 109).