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Part of 1993 USAMO

Problems(1)

Prove that f(x) <=cx

Source:

7/9/2005
Consider functions  f:[0,1]→R \, f: [0,1] \rightarrow \mathbb{R} \, which satisfy (i) f(x)≥0 f(x) \geq 0 \, for all  x \, x \, in  [0,1],\, [0,1], (ii) f(1)=1,f(1) = 1, (iii) f(x)+f(y)≤f(x+y) f(x) + f(y) \leq f(x+y)\, whenever  x, y, \, x, \, y, \, and  x+y \, x + y \, are all in  [0,1]\, [0,1].
Find, with proof, the smallest constant  c \, c \, such that f(x)≤cx f(x) \leq cx for every function  f \, f \, satisfying (i)-(iii) and every  x \, x \, in  [0,1]\, [0,1].
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