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Romania NMO 2022 Grade 12 P1

Source: Romania National Olympiad 2022

April 20, 2022
functionIntegralromaniacalculus

Problem Statement

Let F\mathcal{F} be the set of functions f:R→Rf:\mathbb{R}\to\mathbb{R} such that f(2x)=f(x)f(2x)=f(x) for all x∈R.x\in\mathbb{R}.
[*]Determine all functions f∈Ff\in\mathcal{F} which admit antiderivatives on R.\mathbb{R}. [*]Give an example of a non-constant function f∈Ff\in\mathcal{F} which is integrable on any interval [a,b]⊂R[a,b]\subset\mathbb{R} and satisfies ∫abf(x) dx=0\int_a^bf(x) \ dx=0for all real numbers aa and b.b. Mihai Piticari and Sorin Rădulescu
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