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find interval where integral is half

Source: miklos schweitzer 1995 q2

October 2, 2021
real analysiscalculusintegrationMiklos Schweitzercollege contests

Problem Statement

Given f,g∈L1[0,1]f,g\in L^1[0,1] and ∫01f=∫01g=1\int_0^1 f = \int_0^1 g=1, prove that there exists an interval I st ∫If=∫Ig=12\int_I f = \int_I g=\frac12.
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