HMMT 二月 2006 · GEN2 赛 · 第 3 题

HMMT February 2006 — GEN2 Round — Problem 3

Let C be the unit circle. Four distinct, smaller congruent circles C , C , C , C are internally tangent 1 2 3 4 to C such that C is externally tangent to C and C for i = 1 , . . . , 4 where C denotes C and C i i − 1 i +1 5 1 0 represents C . Compute the radius of C . 4 1

解析

Let C be the unit circle. Four distinct, smaller congruent circles C , C , C , C are 1 2 3 4 internally tangent to C such that C is externally tangent to C and C for i = i i − 1 i +1 1 , . . . , 4 where C denotes C and C represents C . Compute the radius of C . 5 1 0 4 1 √ Answer: 2 − 1 ′ Solution: Let O and O be the centers of C and C respectively, and let C be 1 1 ′ tangent to C, C , C at P , Q , and R respectively. Observe that QORO is a square and 2 4 √ ′ ′ that P, O , and O are collinear. Thus, if r is the desired radius, 1 = r + OO = r + r 2, √ 1 √ so that r = = 2 − 1. 2+1