Problem 2

Let

\mathbb{R}^+

denote the set of positive real numbers. Find all functions

f:\mathbb{R}^+\to\mathbb{R}^+

such that for each

x\in\mathbb{R}^+

, there is exactly one

y\in\mathbb{R}^+

satisfying

xf(y)+yf(x)\le2

Step 4 of 6: Restate the condition after ruling out other friends

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xf(x)\le1\ \text{and}\ xf(y)+yf(x)>2\ \text{for}\ x\ne y

Detailed analysis

Since $x$ is its own unique friend, $xf(x)+xf(x)\le2$ gives $xf(x)\le1$ , and for every $y\ne x$ we must have $xf(y)+yf(x)>2$ (otherwise $y$ would be a second friend of $x$ ).