# IMO 2026 · Problem 5

Let $\mathbb{R}_{>0}$ be the set of positive real numbers. Determine all functions $f\colon \mathbb{R}_{>0}\to\mathbb{R}_{>0}$ such that
$$\sqrt{\frac{x^2+f(y)^2}{2}} \;\ge\; \frac{f(x)+y}{2} \;\ge\; \sqrt{x\,f(y)}$$
for every $x,y\in\mathbb{R}_{>0}$.