If $f(f(x)) = x $ has at most 1 solution, then so does $f(x) = x$. 
Let f be defined on [0,1] and its values are between 0 and 1. If
  $f(f(x)) = x $ has at most 1 solution, then $f(x) = x$ has at most 1
  solution.

Please, give me a hint how to prove this.

show that every solution of $f(x) = x$ is also a solution of $f(f(x)) = x$ which is in fact rather obvious.
