USAMO 1987 Problem 3

$X$ is the smallest set of polynomials $p(x)$ such that: (1) $p(x)=x$ belongs to $X$; and (2) if $r(x)$ belongs to $X$, then $x r(x)$ and $(x+(1-x) r(x))$ both belong to $X$. Show that if $r(x)$ and $s(x)$ are distinct elements of $X$, then $\mathrm{r}(\mathrm{x}) \neq \mathrm{s}(\mathrm{x})$ for any $0<\mathrm{x}<1$.