PUMaC 2014 · 团队赛 · 第 14 题

PUMaC 2014 — Team Round — Problem 14

[ 9 ] Define function f ( x ) (where k is a positive integer) as follows: k k k k f ( x ) = (cos kx )(cos x ) + (sin kx )(sin x ) (cos 2 x ) k Find the sum of all distinct value(s) of k such that f ( x ) is a constant function. k

解析

[ 9 ] Define function f ( x ) (where k is a positive integer) as follows: k k k k f ( x ) = (cos kx )(cos x ) + (sin kx )(sin x ) − (cos 2 x ) k Find the sum of all distinct value(s) of k such that f ( x ) is a constant function. k Solution : Since f (0) = 0, if f is a constant function, then it must be identically equal to 0. Furthermore, k k π kπ k it must be true that f ( ) = sin − ( − 1) = 0. It follows that k ≡ 3 (mod 4). Let k 2 2 π 2 π 4 n − 1 4 n − 1 k = 4 n − 1. Then, f ( x ) = − cos ( ) − cos ( ) = 0. From this, one can obtain k 4 n − 1 4 n − 1 π 2 π π 2 π π 1 cos +cos = 0, (2 cos − 1)(cos +1) = 0, and cos = . So, it is necessary 4 n − 1 4 n − 1 4 n − 1 4 n − 1 4 n − 1 2 that n = 1, which means it is necessary that k = 3. And to show that this is sufficient, check that f ( x ) = 0. So the answer is 3 . 3