HMMT 二月 2018 · ALGNT 赛 · 第 6 题

HMMT February 2018 — ALGNT Round — Problem 6

Let α , β , and γ be three real numbers. Suppose that cos α + cos β + cos γ = 1 sin α + sin β + sin γ = 1 . Find the smallest possible value of cos α .

解析

Let α , β , and γ be three real numbers. Suppose that cos α + cos β + cos γ = 1 sin α + sin β + sin γ = 1 . Find the smallest possible value of cos α . Proposed by: Henrik Boecken √ − 1 − 7 Answer: 4 Let a = cos α + i sin α , b = cosβ + i sin β , and c = cos γ + i sin γ . We then have a + b + c = 1 + i where a , b , c are complex numbers on the unit circle. Now, to minimize cos α = Re[ a ], consider a triangle with vertices a , 1 + i , and the origin. We want a as far away from 1 + i as possible while maintaining a nonnegative imaginary part. This is achieved when b and c have the same argument, so √ | b + c | = | 1 + i − a | = 2. Now a , 0, and 1 + i form a 1 − 2 − 2 triangle. The value of cos α is now √ π the cosine of the angle between the 1 and 2 sides plus the angle from 1 + i . Call the first angle δ . 4 Then √ 2 2 2 1 + ( 2) − 2 √ cos δ = 2 · 1 · 2 − 1 √ = 2 2 and ( ) π cos α = cos + δ 4 π π = cos cos δ − sin sin δ 4 4 √ √ √ 2 − 1 2 7 √ √ = · − · 2 2 2 2 2 2 √ − 1 − 7 = 4