# 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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