Valid Sets (The actual NIMO 2012 3.9)

Probability Level 5

Let f(x)=x22xf(x) = x^2 - 2x. A set of real numbers SS is valid if it satisfies the following:

1) If xSx \in S, then f(x)Sf(x) \in S.
2) If xSx \in S and f(f(fk f’s(x)))=x\underbrace{f(f(\dots f}_{k\ f\text{'s}}(x)\dots )) = x for some integer kk, then f(x)=xf(x) = x.

Find the number of 8-element valid sets.

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