HMMT 二月 2006 · GEN2 赛 · 第 3 题

英文解析

3. 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