HMMT 十一月 2025 · 团队赛 · 第 3 题

HMMT November 2025 — Team Round — Problem 3

[30] Let ABCD and CEF G be squares such that C lies on segment DG and E lies on segment BC . Let O be the circumcenter of triangle AEG . Given that A , D , and O are collinear and AB = 1, compute F G .

解析

[30] Let ABCD and CEF G be squares such that C lies on segment DG and E lies on segment BC . Let O be the circumcenter of triangle AEG . Given that A , D , and O are collinear and AB = 1, compute F G . Proposed by: Sarunyu Thongjarast √ Answer: 3 − 1 Solution: A B E F G D C O Note that O is uniquely determined as the intersection of AD and the perpendicular bisector of GE . ◦ Since CF is the perpendicular bisector of GE , O must lie on CF , meaning ∠ DCO = ∠ F CG = 45 . ◦ ◦ ◦ △ ODC is now a 45 -45 -90 triangle, so OD = DC , and the radius of the circle is AD + OD = 2. As a result, p p √ 2 2 2 2 F G = CG = DG − DC = OG − OD − DC = 2 − 1 − 1 = 3 − 1 . © 2025 HMMT