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Romania NMO 2023 Grade 12 P3

Source: Romania National Olympiad 2023

April 14, 2023
real analysisintegrationfunctionriemann integral

Problem Statement

Let a,b∈Ra,b \in \mathbb{R} with a<b,a < b, 2 real numbers. We say that f:[a,b]→Rf: [a,b] \rightarrow \mathbb{R} has property (P)(P) if there is an integrable function on [a,b][a,b] with property that
f(x)−f(x+a2)=f(x+b2)−f(x),∀x∈[a,b]. f(x) - f \left( \frac{x + a}{2} \right) = f \left( \frac{x + b}{2} \right) - f(x) , \forall x \in [a,b].
Show that for all real number tt there exist a unique function f:[a,b]→Rf:[a,b] \rightarrow \mathbb{R} with property (P),(P), such that ∫abf(x)dx=t.\int_{a}^{b} f(x) \text{dx} = t.
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