HMMT 二月 2009 · 冲刺赛 · 第 22 题

HMMT February 2009 — Guts Round — Problem 22

[ 10 ] A circle having radius r centered at point N is externally tangent to a circle of radius r centered 1 2 at M . Let l and j be the two common external tangent lines to the two circles. A circle centered at P with radius r is externally tangent to circle N at the point at which l coincides with circle N , and 2 line k is externally tangent to P and N such that points M , N , and P all lie on the same side of k . For what ratio r /r are j and k parallel? 1 2 6 4 2

解析

[ 10 ] A circle having radius r centered at point N is tangent to a circle of radius r centered at M . 1 2 Let l and j be the two common external tangent lines to the two circles. A circle centered at P with radius r is externally tangent to circle N at the point at which l coincides with circle N , and line k is 2 externally tangent to P and N such that points M , N , and P all lie on the same side of k . For what ratio r /r are j and k parallel? 1 2 Answer: 3 Solution: Suppose the lines are parallel. Draw the other tangent line to N and P - since M and P have the same radius, it is tangent to all three circles. Let j and k meet circle N at A and B , ◦ respectively. Then by symmetry we see that ∠ AN M = ∠ M N P = ∠ P N B = 60 since A , N , and B are collinear (perpendicular to j and k ). Let D be the foot of the perpendicular from M to AN . In 4 M DN , we have M N = 2 DN , so r + r = 2( r − r ), and so r /r = 3. 1 2 1 2 1 2 6 4 2