HMMT 十一月 2019 · GEN 赛 · 第 9 题

HMMT November 2019 — GEN Round — Problem 9

Let ABCD be an isosceles trapezoid with AD = BC = 255 and AB = 128. Let M be the midpoint of ◦ CD and let N be the foot of the perpendicular from A to CD . If ∠ M BC = 90 , compute tan ∠ N BM .

解析

Let ABCD be an isosceles trapezoid with AD = BC = 255 and AB = 128. Let M be the midpoint of ◦ CD and let N be the foot of the perpendicular from A to CD . If ∠ M BC = 90 , compute tan ∠ N BM . Proposed by: Milan Haiman 120 Answer: 353 Construct P , the reflection of A over CD . Note that P , M , and B are collinear. As ∠ P N C = ∠ P BC = AN ◦ 90 , P N BC is cyclic. Thus, ∠ N BM = ∠ N CP , so our desired tangent is tan ∠ ACN = . Note that CN 1 N M = AB = 64. Since 4 AN D ∼ 4 M AD , 2 255 N D = . 64 + N D 255 AN 120 120 Solving, we find N D = 225, which gives AN = 120. Then we calculate = = . CN 128+225 353