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 \(f_k\big(f_k(\alpha)\big)=\alpha,\) 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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