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Found problems: 4776

2019 District Olympiad, 2
Tags: integrability , function , calculus
Let $n$ be a positive integer and $f:[0,1] \to \mathbb{R}$ be an integrable function. Prove that there exists a point $c \in \left[0,1- \frac{1}{n} \right],$ such that [center] $ \int\limits_c^{c+\frac{1}{n}}f(x)\mathrm{d}x=0$ or $\int\limits_0^c f(x) \mathrm{d}x=\int\limits_{c+\frac{1}{n}}^1f(x)\mathrm{d}x.$ [/center]