HMMT 二月 2023 · 冲刺赛 · 第 28 题

HMMT February 2023 — Guts Round — Problem 28

[20] Suppose ABCD is a convex quadrilateral with ∠ ABD = 105 , ∠ ADB = 15 , AC = 7, and BC = CD = 5. Compute the sum of all possible values of BD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2023, February 18, 2023 — GUTS ROUND Organization Team Team ID#

解析

[20] Suppose ABCD is a convex quadrilateral with ∠ ABD = 105 , ∠ ADB = 15 , AC = 7, and BC = CD = 5. Compute the sum of all possible values of BD . Proposed by: Luke Robitaille √ Answer: 291 ◦ Solution: Let O be the cirumcenter of triangle ABD . By the inscribed angle theorem, ∠ AOC = 90 ◦ and ∠ BOC = 60 . Let AO = BO = CO = x and CO = y . By the Pythagorean theorem on triangle AOC , 2 2 x + y = 49 and by the Law of Cosines on triangle BOC , 2 2 x − xy + y = 25 . √ It suffices to find the sum of all possible values of BD = 3 x . Since the two conditions on x and y are both symmetric, the answer is equal to √ √ √ 2 2 2 2 3( x + y ) = 9( x + y ) − 6( x − xy + y ) = 291 . It is easy to check that both solutions generate valid configurations.