2007 IMO Shortlist Problems/A4

Problem

(Thailand) Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $f(x+f(y)) = f(x+y) + f(y)$ for all $x,y \in \mathbb{R}^+$ . (Symbol $\mathbb{R}^+$ denotes the set of all positive real numbers [sic].)

Solution

We will show that $f(x) = 2x$ is the unique solution to this equation. To this end, let $g(x) = f(x) - x$ .

Lemma 1. The function $g$ is injective.

Proof. Suppose $g(a) = g(b)$ . Then $a = g(b+a+g(a)) - g(b+a) = g(a+b+g(b)) - g(a+b) = b,$ as desired. $\blacksquare$

Lemma 2. If $a>b$ , then $g(a+g(b)) = g(a) + b$ .

Proof. Set $y= b$ , $x=a-b$ . $\blacksquare$

Lemma 3. For all $a,b$ , $g(a+b) = g(a) + g(b)$ .

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2007 IMO Shortlist Problems/A4

Problem

Solution