RO-TST 1978

For any set $A$ we say that two functions $f,g:A\longrightarrow A$ are similar, if there exists a bijection $h:A\longrightarrow A$ such that $f\circ h=h\circ g.$ a) If $A$ has three elements, construct a finite, arbitrary number functions, having as domain and codomain $A,$ that are two by two similar, and every other function with the same domain and codomain as the ones determined is similar to, at least, one of them.b) For $A=\mathbb{R} ,$ show that the functions $\sin$ and $-\sin$ are similar.

A sequence $\left( x_n\right)_{n\ge 0}$ of real numbers satisfies $x_0>1=x_{n+1}\left( x_n-\left\lfloor x_n\right\rfloor\right) ,$ for each $n\ge 1.$ Prove that if $\left( x_n\right)_{n\ge 0}$ is periodic, then $x_0$ is a root of a quadratic equation. Study the converse.