HMMT 二月 2020 · 冲刺赛 · 第 20 题

HMMT February 2020 — Guts Round — Problem 20

[10] There exist several solutions to the equation sin x sin 3 x 1 + = , sin 4 x sin 2 x ◦ ◦ where x is expressed in degrees and 0 < x < 180 . Find the sum of all such solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2020, February 15, 2020 — GUTS ROUND Organization Team Team ID#

解析

[10] There exist several solutions to the equation sin x sin 3 x 1 + = , sin 4 x sin 2 x ◦ ◦ where x is expressed in degrees and 0 < x < 180 . Find the sum of all such solutions. Proposed by: Benjamin Qi ◦ Answer: 320 Solution: We first apply sum-to-product and product-to-sum: sin 4 x + sin x sin 3 x = sin 4 x sin 2 x 2 sin(2 . 5 x ) cos(1 . 5 x ) sin(2 x ) = sin(4 x ) sin(3 x ) Factoring out sin(2 x ) = 0 , sin(2 . 5 x ) cos(1 . 5 x ) = cos(2 x ) sin(3 x ) ◦ Factoring out cos(1 . 5 x ) = 0 (which gives us 60 as a solution), sin(2 . 5 x ) = 2 cos(2 x ) sin(1 . 5 x ) sin(2 . 5 x ) = sin(3 . 5 x ) − sin(0 . 5 x ) Convert into complex numbers, we get