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Problem 5. Let $\mathbb{R}_{ > 0}$ be the set of positive real numbers. Determine all functions $f: \mathbb{R}_{ > 0} \to \mathbb{R}_{ > 0}$ such that
$$\sqrt{\frac{x^2 + f(y)^2}{2}} \geq \frac{f(x) + y}{2} \geq \sqrt{x f(y)}$$
for every $x, y \in \mathbb{R}_{ > 0}$.
Solution 1Solution 2Solution 3Solution 4Solution 5
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