This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 1687

2010 VJIMC, Problem 3
Tags: integration , calculus
Prove that there exist positive constants $c_1$ and $c_2$ with the following properties: a) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\cos(kx)\text dx\right|<\frac{c_1}{k^{3/2}}.$$b) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\sin(kx)\text dx\right|<\frac{c_2}k.$$

2009 Today's Calculation Of Integral, 463
Tags: calculus computations , trigonometry , integration , calculus
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{e^{\frac{1}{\cos ^ 2 x}}\sin x}{\cos ^ 3 x}\ dx$.

2021 CMIMC Integration Bee, 7
Tags: integration , calculus
$$\int_0^\infty \frac{1}{(x^2+4)^{5/2}}\,dx$$ [i]Proposed by Connor Gordon[/i]

2001 IMC, 3
Tags: limit , real analysis , integration
Find $\lim_{t\rightarrow 1^-} (1-t) \sum_{n=1}^{\infty}\frac{t^n}{1+t^n}$.

2010 Today's Calculation Of Integral, 568
Tags: limit , probability , calculus , integration , calculus computations , logarithm
Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$

2021 CMIMC Integration Bee, 8
Tags: integration , calculus
$$\int\left(\frac{x-1}{x^2+1}\right)^2e^x\,dx$$ [i]Proposed by Connor Gordon[/i]

2000 Tuymaada Olympiad, 3
Tags: calculus , integration , combinatorics
Can the 'brick wall' (infinite in all directions) drawn at the picture be made of wires of length $1, 2, 3, \dots$ (each positive integral length occurs exactly once)? (Wires can be bent but should not overlap; size of a 'brick' is $1\times 2$). [asy] unitsize(0.5 cm); for(int i = 1; i <= 9; ++i) { draw((0,i)--(10,i)); } for(int i = 0; i <= 4; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 1,2*j)--(2*i + 1,2*j + 1)); } } for(int i = 0; i <= 3; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 2,2*j + 1)--(2*i + 2,2*j + 2)); } } [/asy]

2012 Today's Calculation Of Integral, 810
Tags: function , calculus computations , integration , calculus
Given the functions $f(x)=xe^{x}+2x\int_0^2 |g(t)|dt-1,\ g(x)=x^2-x\int_0^1 f(t)dt$, evaluate $\int_0^2 |g(t)|dt.$

2010 Princeton University Math Competition, 6
Tags: algebra , integration , quadratic , quadratic formula , calculus
Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

2009 Romania National Olympiad, 1
Tags: calculus , integration , real analysis , inequalities , definite integration , find all functions , function
Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and $$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$

2023 CMIMC Integration Bee, 7
Tags: integration , calculus
\[\int_0^{\frac \pi 2} \left(\frac{1}{1-\cos(x)}-\frac{2}{x^2}\right)\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2008 Alexandru Myller, 1
Tags: integration , squeeze theorem , calculus
$ \lim_{n\to\infty} n2^n\int_1^n \frac{dx}{\left( 1+x^2\right)^n} $ [i][i]Bogdan Enescu[/i][/i]

2005 Today's Calculation Of Integral, 75
Tags: calculus computations , integration , function , trigonometry , induction , calculus
A function $f(\theta)$ satisfies the following conditions $(a),(b)$. $(a)\ f(\theta)\geq 0$ $(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1$ Prove the following inequality. \[\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)\]

2011 Today's Calculation Of Integral, 724
Tags: logarithm , integration , calculus , limit , calculus computations
Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.

2007 AIME Problems, 8
Tags: calculus , integration , derivative , modular arithmetic , rectangle , geometry , quadratic
A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.

2012 Today's Calculation Of Integral, 827
Tags: calculus , trigonometry , integration , limit , calculus computations
Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$

2005 ISI B.Stat Entrance Exam, 2
Tags: algebra , polynomial , integration , function , derivative , calculus
Let \[f(x)=\int_0^1 |t-x|t \, dt\] for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?

2005 Today's Calculation Of Integral, 67
Tags: calculus computations , integration , calculus , geometry
Evaluate \[\frac{2005\displaystyle \int_0^{1002}\frac{dx}{\sqrt{1002^2-x^2}+\sqrt{1003^2-x^2}}+\int_{1002}^{1003}\sqrt{1003^2-x^2}dx}{\displaystyle \int_0^1\sqrt{1-x^2}dx}\]

2007 Today's Calculation Of Integral, 170
Tags: algebra , calculus computations , calculus , integration , partial fractions , logarithm
Let $a,\ b$ be constant numbers such that $a^{2}\geq b.$ Find the following definite integrals. (1) $I=\int \frac{dx}{x^{2}+2ax+b}$ (2) $J=\int \frac{dx}{(x^{2}+2ax+b)^{2}}$

2012 Today's Calculation Of Integral, 786
Tags: integration , limit , algebra , probability , calculus , induction , polynomial
For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$ (1) Find $H_1(x),\ H_2(x),\ H_3(x)$. (2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction. (3) Let $a$ be real number. For $n\geq 3$, express $S_n(a)=\int_0^a xH_n(x)e^{-x^2}dx$ in terms of $H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0)$. (4) Find $\lim_{a\to\infty} S_6(a)$. If necessary, you may use $\lim_{x\to\infty}x^ke^{-x^2}=0$ for a positive integer $k$.

2005 Today's Calculation Of Integral, 53
Tags: calculus , integration , trigonometry , calculus computations
Find the maximum value of the following integral. \[\int_0^{\infty} e^{-x}\sin tx\ dx\]

2007 Today's Calculation Of Integral, 218
Tags: function , integration , calculus , quadratic , trigonometry , calculus computations
For any quadratic functions $ f(x)$ such that $ f'(2)\equal{}1$, evaluate $ \int_{2\minus{}\pi}^{2\plus{}\pi}f(x)\sin\left(\frac{x}{2}\minus{}1\right) dx$.

2010 Today's Calculation Of Integral, 604
Tags: integration , limit , calculus , calculus computations
Let $r$ be a positive integer. Determine the value of $a$ for which the limit value $\lim_{n\to\infty} \frac{\sum_{k=1}^n k^r}{n^a} $ has a non zero finite value, then find the limit value. 1956 Tokyo Institute of Technology entrance exam

2010 Putnam, B5
Tags: integration , limit , logarithm , inequalities , induction , function
Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$

2007 ITest, 31
Tags: geometry , integration , calculus , inequalities
Let $x$ be the length of one side of a triangle and let $y$ be the height to that side. If $x+y=418$, find the maximum possible $\textit{integral value}$ of the area of the triangle.