Olympiad problems

2023 Romania National Olympiad, 3

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

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2011 Bogdan Stan, 3

Find all Riemann integrable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ which have the property that, for all nonconstant and continuous functions $ g:\mathbb{R}\longrightarrow\mathbb{R}, $ and all real numbers $ a,b $ such that $ a<b, $ the following equality holds. $$ \int_a^b \left( f\circ g \right) (x)dx=\int_a^b \left( g\circ f \right) (x)dx $$ [i]Cosmin Nițu[/i]

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2004 District Olympiad, 2

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous function such that $$ \int_0^1 f(x)g(x)dx =\int_0^1 f(x)dx\cdot\int_0^1 g(x)dx , $$ for all functions $ g:[0,1]\longrightarrow\mathbb{R} $ that are continuous and non-differentiable. Prove that $ f $ is constant.

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