HMMT 二月 2025 · 冲刺赛 · 第 25 题

HMMT February 2025 — Guts Round — Problem 25

[14] Let ABCD be a trapezoid such that AB ∥ CD , AD = 13, BC = 15, AB = 20, and CD = 34. Point X lies inside the trapezoid such that ∠ XAB = 2 ∠ XBA and ∠ XDC = 2 ∠ XCD . Compute XD − XA .

解析

[14] Let ABCD be a trapezoid such that AB ∥ CD , AD = 13, BC = 15, AB = 20, and CD = 34. Point X lies inside the trapezoid such that ∠ XAB = 2 ∠ XBA and ∠ XDC = 2 ∠ XCD . Compute XD − XA . Proposed by: Pitchayut Saengrungkongka Answer: 4 Solution: P B A X C D Q Construct point P on AB such that XA = XP and point Q on CD such that XD = XQ . The angle condition gives QC = XQ = XD and P B = XP = XA . Moreover, ADQP is an isosceles trapezoid. Let S be the projection of A onto CD , and let T be on CD such that AT ∥ BC . Then ADT is a 13-14-15 triangle, so DS = 5. Therefore, QD − P A = 10. Finally, we get XD − XA = QC − P B = (34 − QD ) − (20 − P A ) = 14 − 10 = 4 .