HMMT 十一月 2025 · 冲刺赛 · 第 7 题

HMMT November 2025 — Guts Round — Problem 7

[7] Point X lies on diagonal AC of rectangle ABCD such that AX = 11, CX = 1, and triangle BXD has area 18. Given that BX < DX , compute BX .

解析

[7] Point X lies on diagonal AC of rectangle ABCD such that AX = 11, CX = 1, and triangle BXD has area 18. Given that BX < DX , compute BX . Proposed by: Jason Mao √ Answer: 13 Solution: Let AC and BD meet at M , and let T be the foot of the altitude from X onto BD . A B T M X D C Diagonals AC and BD both have length AX + CX = 12. Since △ BXD has area 18, we have: 1 18 · 2 ( BD )( T X ) = 18 = ⇒ T X = = 3 . 2 12 Furthermore, M X = M C − CX = 6 − 1 = 5. Then the Pythagorean Theorem on △ M T X yields: p p 2 2 2 2 M T = M X − T X = 5 − 3 = 4 . Finally, BT = M B − M T = 6 − 4 = 2, so the Pythagorean Theorem on △ BT X yields: p p √ 2 2 2 2 BX = T B + T X = 2 + 3 = 13 .