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Source: Romanian National Olympiad 2024 - Grade 12 - Problem 1

April 3, 2024
real analysiscalculusintegrationfunction

Problem Statement

Let f:R→Rf: \mathbb{R} \to \mathbb{R} be a continuous function such that f(x)+sin⁡(f(x))≥x,f(x)+\sin(f(x)) \ge x, for all x∈R.x \in \mathbb{R}. Prove that ∫0πf(x)dx≥π22−2.\int\limits_0^{\pi} f(x) \mathrm{d}x \ge \frac{\pi^2}{2}-2.
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