PUMaC 2016 · 几何（B 组） · 第 1 题

PUMaC 2016 — Geometry (Division B) — Problem 1

A circle of radius 1 has four circles ω , ω , ω , and ω of equal radius internally tangent to it, 1 2 3 4 so that ω is tangent to ω , which is tangent to ω , which is tangent to ω , which is tangent 1 2 3 4 to ω , as shown. The radius of the circle externally tangent to ω , ω , ω , and ω has radius 1 1 2 3 4 √ r . If r = a − b for positive integers a and b , compute a + b .

解析

Let s be the radius of each of the ω . Then we can write r + 2 s = 1 since the radius of the big i circle is the diameter of ω (for example) plus the radius of the circle in the middle. We can 1 also draw an isosceles right triangle with vertices at the center of the middle circle, the center √ of ω , and the point of tangency of ω and ω . This triangle gives us r + s = s 2. Subtracting 1 1 2 √ the two equations, we get s = 1 − s 2. We sovle this: √ s = 1 − s 2 √ s (1 + 2) = 1 √ 1 √ s = = 2 − 1 . 1 + 2 √ √ Thus, r = 1 − 2 s = 3 − 2 2 = 3 − 8. Our answer is thus 3 + 8 = 11 . Problem written by Eric Neyman.