CGMO 2017 Problem 6

Given a finite set $X$ , two positive integers $n,k$ , and a map $f:X\to X$ . Define $f^{(1)}(x)=f(x),f^{(i+1)}(x)=f^{(i)}(x)$ , $i=1,2,3,\ldots$ . It is known that for any $x\in X$ , $f^{(n)}(x)=x$ . Define $m_j$ the number of $x\in X$ satisfying $f^{(j)}(x)=x$ . Prove that: (1) $\frac{1}n \sum_{j=1}^n m_j\sin {\frac{2kj\pi}{n}}=0$ (2) $\frac{1}n \sum_{j=1}^n m_j\cos {\frac{2kj\pi}{n}}$ is a non-negative integer.