HMMT 十一月 2022 · GEN 赛 · 第 7 题

HMMT November 2022 — GEN Round — Problem 7

In circle ω , two perpendicular chords intersect at a point P. The two chords have midpoints M and 1 M respectively, such that P M = 15 and P M = 20 . Line M M intersects ω at points A and B , 2 1 2 1 2 with M between A and M . Compute the largest possible value of BM − AM . 1 2 2 1

解析

In circle ω , two perpendicular chords intersect at a point P. The two chords have midpoints M and 1 M respectively, such that P M = 15 and P M = 20 . Line M M intersects ω at points A and B , 2 1 2 1 2 with M between A and M . Compute the largest possible value of BM − AM . 1 2 2 1 Proposed by: Vidur Jasuja Answer: 7 Solution: Let O be the center of ω and let M be the midpoint of AB (so M is the foot of O to M M ). 1 2 Since OM P M is a rectangle, we easily get that M M = 16 and M M = 9. Thus, BM − AM = 1 2 1 2 2 1 M M − M M = 7. 1 2