HMMT 十一月 2023 · 团队赛 · 第 1 题

HMMT November 2023 — Team Round — Problem 1

[20] Let ABC be an equilateral triangle with side length 2 that is inscribed in a circle ω . A chord of ω passes through the midpoints of sides AB and AC . Compute the length of this chord. x x 1 /x 1 /x

解析

[20] Let ABC be an equilateral triangle with side length 2 that is inscribed in a circle ω . A chord of ω passes through the midpoints of sides AB and AC . Compute the length of this chord. Proposed by: Rishabh Das √ Answer: 5 A M T N X Y O B C Solution 1: Let O and r be the center and the circumradius of △ ABC . Let T be the midpoint of the chord in question. √ AB 2 3 √ Note that AO = = . Additionally, we have that AT is half the distance from A to BC , i.e. 3 3 √ √ 3 3 AT = . This means that T O = AO − AT = . By the Pythagorean Theorem, the length of the 2 6 chord is equal to: r r p √ 4 1 5 2 2 2 r − OT = 2 − = 2 = 5 . 3 12 4 Solution 2: Let the chord be XY , and the midpoints of AB and AC be M and N , respectively, so that the chord has points X, M, N, Y in that order. Let XM = N Y = x . Power of a point gives √ − 1 ± 5 2 1 = x ( x + 1) = ⇒ x = . 2 √ Taking the positive solution, we have XY = 2 x + 1 = 5 . x x 1 /x 1 /x