Olympiad problems

2021 Science ON grade XII, 3

Define $E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$ such that $E$ posseses the following properties:\\ $\textbf{(i)}$ If $\int_0^1 f(x)g(x) dx = 0$ for $f\in E$ with $\int_0^1f^2(t)dt \neq 0$, then $g\in E$; \\ $\textbf{(ii)}$ There exists $h\in E$ with $\int_0^1 h^2(t)dt\neq 0$.\\ Prove that $E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$. \\ [i](Andrei Bâra)[/i]

2011 Bogdan Stan, 3

Tags: calculus , integrability , riemann integral , integration , function

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]

2019 District Olympiad, 2

Tags: function , integrability , calculus

Let $n$ be a positive integer and $f:[0,1] \to \mathbb{R}$ be an integrable function. Prove that there exists a point $c \in \left[0,1- \frac{1}{n} \right],$ such that [center] $ \int\limits_c^{c+\frac{1}{n}}f(x)\mathrm{d}x=0$ or $\int\limits_0^c f(x) \mathrm{d}x=\int\limits_{c+\frac{1}{n}}^1f(x)\mathrm{d}x.$ [/center]

2021 Science ON all problems, 3

Tags: real analysis , integrability , riemann integrability , function

Define $E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$ such that $E$ posseses the following properties:\\ $\textbf{(i)}$ If $\int_0^1 f(x)g(x) dx = 0$ for $f\in E$ with $\int_0^1f^2(t)dt \neq 0$, then $g\in E$; \\ $\textbf{(ii)}$ There exists $h\in E$ with $\int_0^1 h^2(t)dt\neq 0$.\\ Prove that $E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$. \\ [i](Andrei Bâra)[/i]

2011 Bogdan Stan, 4

Tags: calculus , derivative , integrability , convexity , side-continuity , function

Let be an open interval $ I $ and a convex function $ f:I\longrightarrow\mathbb{R} . $ Prove that the lateral derivatives of $ f $ are left-continuous on $ \mathbb{R} $ and also right-continuous on $ \mathbb{R} . $ [i]Marin Tolosi[/i]