HMMT 二月 2022 · 冲刺赛 · 第 28 题

HMMT February 2022 — Guts Round — Problem 28

[14] Compute the nearest integer to ∞ X π n 3 100 3 sin . n 3 n =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2022, February 19, 2022 — GUTS ROUND Organization Team Team ID#

解析

[14] Compute the nearest integer to ∞ X π 3 n 100 3 sin . n 3 n =1 Proposed by: Akash Das Answer: 236 Solution: Note that we have 1 3 3 sin 3 x = 3 sin x − 4 sin x = ⇒ sin x = (3 sin x − sin 3 x ) , 4 which implies that 3 sin x 1 sin x sin 3 x = − . 3 x 4 x 3 x π Substituting x = and simplifying gives us n 3 π π sin sin π 3 π n n − 1 n 3 3 3 3 sin = − . π π n 3 4 n n − 1 3 3 Summing this from n = 1 to N and telescoping gives us N π X sin π 3 π sin π N n 3 3 3 sin = − . π n 3 4 π N 3 n =1 Taking N → ∞ gives us an answer of 3 π 100 · ≈ 236 . 4