USAMO 1993 Problem 3

Let $\mathrm{S}$ be the set of functions $\mathrm{f}$ defined on reals in the closed interval $[0,1]$ with non-negative real values such that $\mathrm{f}(1)=1$ and $\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \leq \mathrm{f}(\mathrm{x}+\mathrm{y})$ for all $\mathrm{x}, \mathrm{y}$ such that $\mathrm{x}+\mathrm{y} \leq 1$. What is the smallest $\mathrm{k}$ such that $\mathrm{f}(\mathrm{x}) \leq$ $\mathrm{kx}$ for all $\mathrm{f}$ in $\mathrm{S}$ and all $\mathrm{x}$ ?