HMMT 二月 2007 · 冲刺赛 · 第 11 题
HMMT February 2007 — Guts Round — Problem 11
题目详情
problem 11 will not alter the correctness of your answer to problem 12.) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUND 2048 n
解析
- [ 8 ] Let A denote the answer to problem 10. Two circles lie in the plane; denote the lengths of the 10 internal and external tangents between these two circles by x and y , respectively. Given that the product of the radii of these two circles is 15 / 2, and that the distance between their centers is A , 10 2 2 determine y − x . Answer: 30 . Suppose the circles have radii r and r . Then using the tangents to build right 1 2 2 2 2 2 2 2 2 2 2 triangles, we have x + ( r + r ) = A = y + ( r − r ) . Thus, y − x = ( r + r ) − ( r − r ) = 1 2 1 2 1 2 1 2 10 4 r r = 30. 1 2