Olympiad problems

2023 CMIMC Integration Bee, 5

Tags: integration , calculus

\[\int_1^\infty \frac{1}{x\sqrt{x^{2023}-1}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2010 Today's Calculation Of Integral, 540

Tags: calculus , calculus computations , logarithm , integration

Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.

2005 Today's Calculation Of Integral, 35

Tags: calculus , derivative , function , integration , calculus computations

Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.

2011 Today's Calculation Of Integral, 739

Tags: functional equation , calculus , algebra , derivative , integration , trigonometry , function

Find the function $f(x)$ such that : \[f(x)=\cos x+\int_0^{2\pi} f(y)\sin (x-y)\ dy\]

2005 Today's Calculation Of Integral, 7

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

Calculate the following indefinite integrals. [1] $\int \sqrt{x}(\sqrt{x}+1)^2 dx$ [2] $\int (e^x+2e^{x+1}-3e^{x+2})dx$ [3] $\int (\sin ^2 x+\cos x)\sin x dx$ [4] $\int x\sqrt{2-x} dx$ [5] $\int x\ln x dx$

2011 Romania National Olympiad, 2

Tags: calculus , integration , function , real analysis

Let be a continuous function $ f:[0,1]\longrightarrow\left( 0,\infty \right) $ having the property that, for any natural number $ n\ge 2, $ there exist $ n-1 $ real numbers $ 0<t_1<t_2<\cdots <t_{n-1}<1, $ such that $$ \int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt. $$ Calculate $ \lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} . $

2005 Today's Calculation Of Integral, 78

Tags: calculus , integration , calculus computations , trigonometry

Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate \[\int_0^1 \sin \alpha x\sin \beta x\ dx\]

1995 IberoAmerican, 3

Tags: floor function , function , calculus , algebra , integration , induction

A function $f: \N\rightarrow\N$ is circular if for every $p\in\N$ there exists $n\in\N,\ n\leq{p}$ such that $f^n(p)=p$ ($f$ composed with itself $n$ times) The function $f$ has repulsion degree $k>0$ if for every $p\in\N$ $f^i(p)\neq{p}$ for every $i=1,2,\dots,\lfloor{kp}\rfloor$. Determine the maximum repulsion degree can have a circular function. [b]Note:[/b] Here $\lfloor{x}\rfloor$ is the integer part of $x$.

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*}

2007 Today's Calculation Of Integral, 242

Tags: geometry , calculus computations , integration , function , calculus

A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$. Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

1952 AMC 12/AHSME, 1

Tags: calculus , geometry , integration

If the radius of a circle is a rational number, its area is given by a number which is: $ \textbf{(A)}\ \text{rational} \qquad\textbf{(B)}\ \text{irrational} \qquad\textbf{(C)}\ \text{integral} \qquad\textbf{(D)}\ \text{a perfect square}$ $ \textbf{(E)}\ \text{none of these}$

2005 Gheorghe Vranceanu, 3

Tags: calculus , function , real analysis , integration , newton-leibniz , ftc , periodic function

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that: $$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$

2007 China Team Selection Test, 3

Tags: integration , polynomial , algebra , function , quadratic , limit , induction

Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.

2023 CMIMC Integration Bee, 6

Tags: calculus , integration

\[\int_0^2 e^x(x^4+8x^3+18x^2+16x+5)\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2007 Mathematics for Its Sake, 3

Tags: integration , primitive , indefinite integral , function

Let be three positive real numbers $ a,b,c, $ a natural number $ n, $ and the functions $ f:\mathbb{R}\longrightarrow\mathbb{R} ,g:(0,\infty )\longrightarrow\mathbb{R} $ defined as: $$ f(x)=\frac{2(n+1)x^n(x^{n+1}-a) +nx^{n+1} +2a^2x+a}{x^{2n+2}-2ax^{n+1} +a^2x^2+a^2} , $$ $$ g(x)=\frac{a+bx^n}{x+cx^{2n+1}} $$ Calculate the antiderivatives of $ f $ and $ g. $ [i]Nicolae Sanda[/i]

2009 Today's Calculation Of Integral, 459

Tags: calculus , limit , logarithm , integration , calculus computations

Find $ \lim_{x\to\infty} \int_{e^{\minus{}x}}^1 \left(\ln \frac{1}{t}\right)^ n\ dt\ (x\geq 0,\ n\equal{}1,\ 2,\ \cdots)$.

2022 VJIMC, 1

Tags: inequalities , calculus , function , integration

Determine whether there exists a differentiable function $f:[0,1]\to\mathbb R$ such that $$f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.$$

2003 Tuymaada Olympiad, 4

Tags: calculus , integration , algebra , polynomial , number theory

Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite. Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$ [i]Proposed by F. Petrov[/i] [hide="For those of you who liked this problem."] Check [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62259]this thread[/url] out.[/hide]

2010 Today's Calculation Of Integral, 617

Tags: calculus , function , integration , calculus computations

Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane. Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ [i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]

1998 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , integration , geometry

Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$. Find the area of $S$.

2007 Today's Calculation Of Integral, 232

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

For $ f(x)\equal{}1\minus{}\sin x$, let $ g(x)\equal{}\int_0^x (x\minus{}t)f(t)\ dt.$ Show that $ g(x\plus{}y)\plus{}g(x\minus{}y)\geq 2g(x)$ for any real numbers $ x,\ y.$

2009 Today's Calculation Of Integral, 470

Tags: conic , calculus , integration , parabola , calculus computations , geometry

Determin integers $ m,\ n\ (m>n>0)$ for which the area of the region bounded by the curve $ y\equal{}x^2\minus{}x$ and the lines $ y\equal{}mx,\ y\equal{}nx$ is $ \frac{37}{6}$.

2009 Today's Calculation Of Integral, 479

Tags: trigonometry , calculus , integration , calculus computations

Let $ a,\ b$ be real constants. Find the minimum value of the definite integral: $ I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.$

1995 AMC 12/AHSME, 23

Tags: calculus , trigonometry , pythagorean theorem , geometry , triangle inequality , integration , inequalities

The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

2010 N.N. Mihăileanu Individual, 1

Tags: calculus , real analysis , inequalities , lipschitz , integral inequalities , definite integral , integration

Let $ m:[0,1]\longrightarrow\mathbb{R} $ be a metric map. [b]a)[/b] Prove that $ -\text{identity} +m $ is continuous and nonincreasing. [b]b)[/b] Show that $ \int_0^1\int_0^x (-t+m(t))dtdx=\int_0^1 (x-1)(x-m(x))dx. $ [b]c)[/b] Demonstrate that $ \int_0^1\int_0^x m(t)dtdx -\frac{1}{2}\int_0^1 m(x)dx\ge -\frac{1}{12} . $ [i]Gabriela Constantinescu[/i] and [i]Nelu Chichirim[/i]