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A Periodic But Not Fixed Point

For every integer k,k, we define a function fkf_k by the formula

fk(x)=100xksinx.f_k(x)=100x-k\sin x.

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