Olympiad problems

2013 Today's Calculation Of Integral, 884

Tags: integral inequality , calculus , trigonometry , integration , calculus computations , inequalities

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

2019 Korea USCM, 6

Tags: integral inequality , real analysis , function , inequalities

A function $f:[0,\infty)\to[0,\infty)$ is integrable and $$\int_0^\infty f(x)^2 dx<\infty,\quad \int_0^\infty xf(x) dx <\infty$$ Prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)$$

2007 IberoAmerican Olympiad For University Students, 3

Tags: real analysis , integral inequality , inequalities , logarithm , function , calculus , integration

Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds: $\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$, where $T$ is the period of $f(x)$.

1985 Traian Lălescu, 2.2

Tags: inequalities , calculus , young s inequality , integral inequality , bijective , integral , real analysis

Let $ a,b,c\in\mathbb{R}_+^*, $ and $ f:[0,a]\longrightarrow [0,b] $ bijective and non-decreasing. Prove that: $$ \frac{1}{b}\int_0^a f^2 (x)dx +\frac{1}{a}\int_0^b \left( f^{-1} (x)\right)^2dx\le ab. $$

2010 Morocco TST, 1

Tags: function , integral inequality , algebra

$f$ is a function twice differentiable on $[0,1]$ and such that $f''$ is continuous. We suppose that : $f(1)-1=f(0)=f'(1)=f'(0)=0$. Prove that there exists $x_0$ on $[0,1]$ such that $|f''(x_0)| \geq 4$

Today's calculation of integrals, 883

Tags: trigonometry , logarithm , calculus computations , integral inequality , integration , calculus

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2014 ISI Entrance Examination, 7

Tags: calculus , integration , real analysis , rearrangement inequality , integral inequality , inequalities

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}

2013 Today's Calculation Of Integral, 883

Tags: trigonometry , calculus computations , integral inequality , logarithm , calculus , integration

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2015 IMC, 7

Tags: integral inequality

Compute $$ \lim_{A\to+\infty}\frac1A\int_1^A A^{\frac1x}\, dx . $$ Proposed by Jan Šustek, University of Ostrava

2015 Romania National Olympiad, 3

Tags: integral calculus , integral inequality , calculus

Let $\mathcal{C}$ be the set of all twice differentiable functions $f:[0,1] \to \mathbb{R}$ with at least two (not necessarily distinct) zeros and $|f''(x)| \le 1,$ for all $x \in [0,1].$ Find the greatest value of the integral $$\int\limits_0^1 |f(x)| \mathrm{d}x$$ when $f$ runs through the set $\mathcal{C},$ as well as the functions that achieve this maximum. [i]Note: A differentiable function $f$ has two zeros in the same point $a$ if $f(a)=f'(a)=0.$[/i]

2009 IMC, 2

Tags: function , integral inequality , inequalities

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a two times differentiable function satisfying $f(0)=1,f^{\prime}(0)=0$ and for all $x\in [0,\infty)$, it satisfies \[ f^{\prime \prime}(x)-5f^{\prime}(x)+6f(x)\ge 0 \] Prove that, for all $x\in [0,\infty)$, \[ f(x)\ge 3e^{2x}-2e^{3x} \]