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Romania District Olympiad 2001 - Grade XII

Source:

March 16, 2011
functionalgebrapolynomialintegrationcalculusreal analysisreal analysis unsolved

Problem Statement

Consider a continuous function f:[0,1]→Rf:[0,1]\rightarrow \mathbb{R} such that for any third degree polynomial function P:[0,1]→[0,1]P:[0,1]\to [0,1], we have
∫01f(P(x))dx=0\int_0^1f(P(x))dx=0
Prove that f(x)=0, (∀)x∈[0,1]f(x)=0,\ (\forall)x\in [0,1].
Mihai Piticari
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