1993 USAMO Problems/Problem 3

Problem 3

Consider functions $f : [0, 1] \rightarrow \Re$ which satisfy

	(i)	$f(x)\ge0$ for all $x$ in $[0, 1]$ ,
	(ii)	$f(1) = 1$ ,
	(iii)	$f(x) + f(y) \le f(x + y)$ whenever $x$ , $y$ , and $x + y$ are all in $[0, 1]$ .

Find, with proof, the smallest constant $c$ such that

$f(x) \le cx$

for every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$ .

Being typed up now-- 07:01 pm EDT 4/22

1993 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions