USAMO
1993
Consider functions $\, f: [0,1] \rightarrow \mathbb{R} \,$ which satisfy
(i) $f(x) \geq 0 \,$ for all $\, x \,$ in $\, [0,1],$
(ii) $f(1) = 1,$
(iii) $f(x) + f(y) \leq f(x+y)\,$ whenever $\, x, \, y, \,$ and $\, x + y \,$ are all in $\, [0,1]$.
Find, with proof, the smallest constant $\, c \,$ such that
\[ f(x) \leq cx \]for every function $\, f \,$ satisfying (i)-(iii) and every $\, x \,$ in $\, [0,1]$.