HMMT 十一月 2025 · 冲刺赛 · 第 3 题

HMMT November 2025 — Guts Round — Problem 3

[5] Square ABCD has side length 45. Points W , X , Y , and Z lie on sides AB , BC , CD , and DA , respectively, such that AW = CY = 20 and BX = DZ = 25. Compute the area of quadrilateral W XY Z . © 2025 HMMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2025, November 08, 2025 — GUTS ROUND Organization Team Team ID# ◦ ◦

解析

[5] Square ABCD has side length 45. Points W , X , Y , and Z lie on sides AB , BC , CD , and DA , respectively, such that AW = CY = 20 and BX = DZ = 25. Compute the area of quadrilateral W XY Z . Proposed by: Derek Liu Answer: 1000 Solution: © 2025 HMMT W A B Z X D C Y Note that AW = AZ = 20 and BW = BX = 25. Therefore, triangles AW Z and BXW are isosceles √ √ right triangles, whence W Z = 20 2 and W X = 25 2. Both W X and W Z make 45-degree angles with AB , so we get that W Z is perpendicular to W X . Similarly, W X is perpendicular to XY and W Z is perpendicular to ZY . Hence, W XY Z is a rectangle. Thus, the area of W XY Z is √ √ W Z · W X = (20 2) · (25 2) = 1000 . ◦ ◦