# A Periodic But Not Fixed Point

**Geometry**Level 5

For every integer \(k\) we define a function \(f_k \) by the formula \[f_k(x)=100x-k\sin( x)\]

What is the smallest positive integer value of \(k,\) such that for some real \(\alpha\), we have \(f_k(f_k(\alpha))=\alpha,\) but \(f_k(\alpha )\neq \alpha\)?

**Details and assumptions**

The function is evaluated in radians. There is no degree symbol in the problem.