Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME Find the answer

For $x$ a real number, let $f(x)=0$ if $x<1$ and $f(x)=2 x-2$ if $x \geq 1$ . How many solutions are there to the equation $f(f(f(f(x))))=x ?$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Certainly 0,2 are fixed points of $f$ and therefore solutions. On the other hand, there can be no solutions for $x<0$ , since $f$ is nonnegative-valued; for $0<x<2$ , we have $0 \leq f(x)<x<2$ (and $f(0)=0$ ), so iteration only produces values below $x$ , and for $x>2, f(x)>x$ , and iteration produces higher values. So there are no other solutions.

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