HMMT 二月 2008 · 冲刺赛 · 第 12 题

HMMT February 2008 — Guts Round — Problem 12

[ 7 ] Suppose we have an (infinite) cone C with apex A and a plane π . The intersection of π and C is an ellipse E with major axis BC , such that B is closer to A than C , and BC = 4, AC = 5, AB = 3. Suppose we inscribe a sphere in each part of C cut up by E with both spheres tangent to E . What is the ratio of the radii of the spheres (smaller to larger)? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND

解析

[ 7 ] Suppose we have an (infinite) cone C with apex A and a plane π . The intersection of π and C is an ellipse E with major axis BC , such that B is closer to A than C , and BC = 4, AC = 5, AB = 3. Suppose we inscribe a sphere in each part of C cut up by E with both spheres tangent to E . What is the ratio of the radii of the spheres (smaller to larger)? 1 Answer: It can be seen that the points of tangency of the spheres with E must lie on its major 3 axis due to symmetry. Hence, we consider the two-dimensional cross-section with plane ABC . Then the two spheres become the incentre and the excentre of the triangle ABC , and we are looking for the ratio of the inradius to the exradius. Let s , r , r denote the semiperimeter, inradius, and exradius a (opposite to A ) of the triangle ABC . We know that the area of ABC can be expressed as both rs and s −| BC | r 1 r ( s − | BC | ), and so = . For the given triangle, s = 6 and a = 4, so the required ratio is . a r s 3 a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND