HMMT 十一月 2025 · GEN 赛 · 第 5 题

HMMT November 2025 — GEN Round — Problem 5

Let A , B , C , and D be points on a line in that order. There exists a point E such that ∠ AED = 120 and triangle BEC is equilateral. Given that BC = 10 and AD = 39, compute | AB − CD | .

解析

Let A , B , C , and D be points on a line in that order. There exists a point E such that ∠ AED = 120 and triangle BEC is equilateral. Given that BC = 10 and AD = 39, compute | AB − CD | . Proposed by: Jackson Dryg Answer: 21 Solution: E A B C D ◦ First, we note that ∠ ABE = ∠ AED = 120 , so △ ABE ∼ △ AED . Similarly, △ ECD ∼ △ AED , so △ ABE ∼ △ ECD . Now, let AB = x and CD = y . Then, x 10 = = ⇒ xy = 100 . 10 y Furthermore, x + y = AD − BC = 29, so 2 2 2 ( x − y ) = ( x + y ) − 4 xy = 29 − 4 · 100 = 441 . √ Hence, the answer is | x − y | = 441 = 21 .