Problem 410
Circle and tangent line
Let C be the circle with radius r, x2 + y2 = r2. We choose two points P(a, b) and Q(-a, c) so that the line passing through P and Q is tangent to C.
For example, the quadruplet (r, a, b, c) = (2, 6, 2, -7) satisfies this property.
Let F(R, X) be the number of the integer quadruplets (r, a, b, c) with this property, and with 0 < r ≤ R and 0 < a ≤ X.
We can verify that F(1, 5) = 10, F(2, 10) = 52 and F(10, 100) = 3384.
Find F(108, 109) + F(109, 108).
圆与切线
记C为半径为r的圆，其方程为x2 + y2 = r2。我们选取两个点P(a, b)和Q(-a, c)，使得过PQ的直线与圆C相切。
例如，四元组(r, a, b, c) = (2, 6, 2, -7)就满足上述性质。
记F(R, X)是满足上述性质的整数四元组(r, a, b, c)的数目，要求0 < r ≤ R且0 < a ≤ X。
可以验证F(1, 5) = 10, F(2, 10) = 52以及F(10, 100) = 3384。
求F(108, 109) + F(109, 108)。