HMMT 二月 2023 · 冲刺赛 · 第 14 题

HMMT February 2023 — Guts Round — Problem 14

[14] Acute triangle ABC has circumcenter O . The bisector of ∠ ABC and the altitude from C to side AB ◦ intersect at X . Suppose that there is a circle passing through B , O , X , and C . If ∠ BAC = n , where n is a positive integer, compute the largest possible value of n .

解析

[14] Acute triangle ABC has circumcenter O . The bisector of ∠ ABC and the altitude from C to side ◦ AB intersect at X . Suppose that there is a circle passing through B , O , X , and C . If ∠ BAC = n , where n is a positive integer, compute the largest possible value of n . Proposed by: Luke Robitaille Answer: 67 ◦ ◦ Solution: We have ∠ XBC = B/ 2 and ∠ XCB = 90 − B . Thus, ∠ BXC = 90 + B/ 2. We have ∠ BOC = 2 A , so ◦ 90 + B/ 2 = 2 A. ◦ ◦ This gives B = 4 A − 180 , which gives C = 360 − 5 A . ◦ ◦ ◦ ◦ ◦ ◦ In order for 0 < B < 90 , we need 45 < A < 67 . 5 . In order for 0 < C < 90 , we require ◦ ◦ ◦ 54 < A < 72 . The largest integer value in degrees satisfying these inequalities is A = 67 .