HMMT 十一月 2020 · 冲刺赛 · 第 22 题

HMMT November 2020 — Guts Round — Problem 22

[12] In triangle ABC , AB = 32, AC = 35, and BC = x . What is the smallest positive integer x such 2 2 2 that 1 + cos A , cos B , and cos C form the sides of a non-degenerate triangle?

解析

[12] In triangle ABC , AB = 32, AC = 35, and BC = x . What is the smallest positive integer x such 2 2 2 that 1 + cos A , cos B , and cos C form the sides of a non-degenerate triangle? Proposed by: Hahn Lheem Answer: 48 2 2 2 Solution: By the triangle inequality, we wish cos B + cos C > 1 + cos A . The other two inequalities 2 2 2 are always satisfied, since 1 + cos A ≥ 1 ≥ cos B, cos C . Rewrite the above as 2 2 2 2 − sin B − sin C > 2 − sin A, 2 2 2 so it is equivalent to sin B +sin C < sin A . By the law of sines, sin A : sin B : sin C = BC : AC : AB . Therefore, 2 2 2 2 2 2 sin B + sin C < sin A ⇐⇒ CA + AB < x . 2 2 2 Since CA + AB = 2249, the smallest possible value of x such that x > 2249 is 48.