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Romanian District Olympiad 2019 - Grade 12 - Problem 2

Source: Romanian District Olympiad 2019 - Grade 12 - Problem 2

March 16, 2019
integrabilityfunctioncalculus

Problem Statement

Let nn be a positive integer and f:[0,1]→Rf:[0,1] \to \mathbb{R} be an integrable function. Prove that there exists a point c∈[0,1−1n],c \in \left[0,1- \frac{1}{n} \right], such that ∫cc+1nf(x)dx=0 \int\limits_c^{c+\frac{1}{n}}f(x)\mathrm{d}x=0 or ∫0cf(x)dx=∫c+1n1f(x)dx.\int\limits_0^c f(x) \mathrm{d}x=\int\limits_{c+\frac{1}{n}}^1f(x)\mathrm{d}x.
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