PUMaC 2011 · 几何（B 组） · 第 5 题

PUMaC 2011 — Geometry (Division B) — Problem 5

[ 5 ] Four circles are situated in the plane so that each is tangent to the other three. If three of the radii are 5, 5, and 8, the largest possible radius of the fourth circle is a/b , where a and b are positive integers and gcd( a, b ) = 1. Find a + b .

解析

Note: The problem was flawed as stated on the exam. Many thanks to Will Zhang of PEA Green for pointing out that there is a configuration of the three circles of given radii that can give rise to arbitrarily large radii for the fourth circle. If the problem were reworded to specify that the three circles with given radii were externally tangent to one another, the following would have been the solution: 2 The largest possible radius of the fourth circle is achieved when it is internally tangent to the first three. Let O and O be the centers of the circles of radius 5 and let O be the center 1 2 3 of the circle of radius 8. Let O be the center of the largest circle. Note that O must be on the altitude O H of the triangle O O O . Let r be the radius of the largest circle, and let 3 1 2 3 θ = ∠ OO O . Note that O HO is a 5-12-13 right triangle, so HO = 12. From this right 3 2 2 3 3 triangle, we find cos θ = 12 / 13. Then, from the theorem of cosines in triangle OO O we find 3 2 that 12 2 2 2 ( r − 8) + 13 − 2( r − 8)13 · = ( r − 5) . 13 Simplifying the above equation yields 2 2 2 2 2 r − 16 r + 8 + 13 − 24( r − 8) = r − 10 r + 5 2 2 2 ⇒ 8 + 13 − 5 + 8 · 24 = 30 r 40 2 2 ⇒ 30 r = 8 + 12 + 8 · 24 = 16(4 + 9 + 12) = 16 · 25 ⇒ r = . 3 Thus the answer is 40 + 3 = 43 . Figure 4: Problem 5 diagram.