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

$$\sqrt{\frac{x^2 + f(y)^2}{2}} \geq \frac{f(x) + y}{2} \geq \sqrt{x f(y)}$$

for every $x, y \in \mathbb{R}_{>0}$.