HMMT 二月 2018 · 冲刺赛 · 第 17 题

HMMT February 2018 — Guts Round — Problem 17

[ 10 ] Compute the value of ◦ ◦ ◦ cos 30 . 5 + cos 31 . 5 + ... + cos 44 . 5 . ◦ ◦ ◦ sin 30 . 5 + sin 31 . 5 + ... + sin 44 . 5

解析

[ 10 ] Compute the value of ◦ ◦ ◦ cos 30 . 5 + cos 31 . 5 + ... + cos 44 . 5 . ◦ ◦ ◦ sin 30 . 5 + sin 31 . 5 + ... + sin 44 . 5 Proposed by: Sujay Kazi √ √ √ √ √ √ Answer: ( 2 − 1)( 3 + 2) = 2 − 2 − 3 + 6 Consider a 360-sided regular polygon with side length 1 , rotated so that its sides are at half-degree ◦ ◦ ◦ inclinations (that is, its sides all have inclinations of 0 . 5 , 1 . 5 , 2 . 5 , and so on. Go to the bottom point on this polygon and then move clockwise, numbering the sides 1 , 2 , 3 , ..., 360 as you go. Then, take the ◦ ◦ ◦ section of 15 sides from side 31 to side 45 . These sides have inclinations of 30 . 5 , 31 . 5 , 32 . 5 , and so on, ◦ up to 44 . 5 . Therefore, over this section, the horizontal and vertical displacements are, respectively: ◦ ◦ ◦ H = cos 30 . 5 + cos 31 . 5 + ... + cos 44 . 5 ◦ ◦ ◦ V = sin 30 . 5 + sin 31 . 5 + ... + sin 44 . 5 However, we can also see that, letting R be the circumradius of this polygon: ◦ ◦ H = R (sin 45 − sin 30 ) ◦ ◦ V = R [(1 − cos 45 ) − (1 − cos 30 )] √ √ √ √ √ √ H From these, we can easily compute that our desired answer is = ( 2 − 1)( 3+ 2) = 2 − 2 − 3+ 6 . V