# 2012-8 Non-fixed points

Let X be a finite non-empty set. Suppose that there is a function $$f:X\to X$$ such that $$f^{20120407}(x)=x$$ for all $$x\in X$$. Prove that the number of elements x in X such that $$f(x)\neq x$$ is divisible by 20120407.

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