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\int\limits_0^\pi f(x)dx=0, \qquad \int\limits_0^\pi f(x)\cos x dx=0

Source: 1979 Swedish Mathematical Competition p4

March 26, 2021
integrationanalysisalgebra

Problem Statement

f(x)f(x) is continuous on the interval [0,π][0, \pi] and satisfies ∫0πf(x)dx=0,∫0πf(x)cos⁡xdx=0 \int\limits_0^\pi f(x)dx=0, \qquad \int\limits_0^\pi f(x)\cos x dx=0 Show that f(x)f(x) has at least two zeros in the interval (0,π)(0, \pi).
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