0%

Problem 510


Problem 510


Tangent Circles

Circles A and B are tangent to each other and to line L at three distinct points.
Circle C is inside the space between A, B and L, and tangent to all three.

Let rA, rB and rC be the radii of A, B and C respectively.

Let S(n) = Σ rA + rB + rC, for 0 < rA ≤ rB ≤ n where rA, rB and rC are integers.

The only solution for 0 < rA ≤ rB ≤ 5 is rA = 4, rB = 4 and rC = 1, so S(5) = 4 + 4 + 1 = 9.

You are also given S(100) = 3072.
Find S(109).


相切的圆

圆A和圆B彼此相切,同时和直线L也分别相切,三个切点互不相同。
圆C在A、B和L之间,且与这三者都相切。

分别记圆A、B和C的半径为rA、rB和rC

对于所有满足0 < rA ≤ rB ≤ n,且rA、rB和rC均为整数的可行解,我们定义S(n) = Σ rA + rB + rC

对于0 < rA ≤ rB ≤ 5,唯一解是rA = 4,rB = 4,rC = 1,所以S(5) = 4 + 4 + 1 = 9。

此外还已知S(100) = 3072。
求S(109)。