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.


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