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

2024 CMIMC Integration Bee, 12
Tags: integration , calculus
\[\int_1^\infty \frac{\sec^{-1}(x^{2})-\sec^{-1}(\sqrt x)}{x\log(x)}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2014 Turkey MO (2nd round), 5
Tags: integration , algebra
Find all natural numbers $n$ for which there exist non-zero and distinct real numbers $a_1, a_2, \ldots, a_n$ satisfying \[ \left\{a_i+\dfrac{(-1)^i}{a_i} \, \Big | \, 1 \leq i \leq n\right\} = \{a_i \mid 1 \leq i \leq n\}. \]

2011 Kazakhstan National Olympiad, 6
Tags: inequalities , integration , algebra , quadratic
Given a positive integer $n$. One of the roots of a quadratic equation $x^{2}-ax +2 n = 0$ is equal to $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}$. Prove that $2\sqrt{2n}\le a\le 3\sqrt{n}$

2005 Taiwan National Olympiad, 3
Tags: calculus , integration , quadratic , algebra
If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

2007 Today's Calculation Of Integral, 179
Tags: calculus , calculus computations , logarithm , integration
Evaluate the following integrals. (1) Meiji University $\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$ (2) Tokyo University of Science $\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$

2010 Today's Calculation Of Integral, 635
Tags: calculus , trigonometry , derivative , integration , function , calculus computations
Suppose that a function $f(x)$ defined in $-1<x<1$ satisfies the following properties (i) , (ii), (iii). (i) $f'(x)$ is continuous. (ii) When $-1<x<0,\ f'(x)<0,\ f'(0)=0$, when $0<x<1,\ f'(x)>0$. (iii) $f(0)=-1$ Let $F(x)=\int_0^x \sqrt{1+\{f'(t)\}^2}dt\ (-1<x<1)$. If $F(\sin \theta)=c\theta\ (c :\text{constant})$ holds for $-\frac{\pi}{2}<\theta <\frac{\pi}{2}$, then find $f(x)$. [i]1975 Waseda University entrance exam/Science and Technology[/i]

2022 CMIMC Integration Bee, 3
Tags: integration , calculus
\[\int_0^1 x\sqrt[4]{1-x}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4
Tags: logarithm , real analysis , calculus , parameterization , integration
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.

2009 Today's Calculation Of Integral, 458
Tags: geometry , calculus , logarithm , vector , integration , calculus computations
Let $ S(t)$ be the area of the traingle $ OAB$ with $ O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t)$. Evaluate $ \int_1^ e S(t)^2\ln t\ dt$.

2007 Today's Calculation Of Integral, 176
Tags: calculus computations , symmetry , integration , trigonometry , calculus , limit , induction
Let $f_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{\sqrt{k(k+1)}}.$ Find $\lim_{n\to\infty}\int_{0}^{2\pi}\{f_{n}(x)\}^{2}dx.$

2022 CMIMC Integration Bee, 14
Tags: integration , calculus
\[\int_2^\infty \frac{\pi(x)}{x^3 - x}\,dx\] [i]Proposed by Vlad Oleksenko[/i]

2002 Romania National Olympiad, 2
Tags: real analysis , integration , inequalities , calculus , function
Let $f:[0,1]\rightarrow\mathbb{R}$ be an integrable function such that: \[0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.\] Show that there exists $x_1\not= x_2, x_1,x_2\in [0,1]$, such that: \[\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}\]

2007 Today's Calculation Of Integral, 198
Tags: trigonometry , integration , calculus , logarithm , calculus computations
Compare the values of the following definite integrals. \[\int_{0}^{\infty}\ln \left(x+\frac{1}{x}\right)\frac{dx}{1+x^{2}},\ \ \int_{0}^{\frac{\pi}{2}}\left(\frac{\theta}{\sin \theta}\right)^{2}d\theta\]

2010 Today's Calculation Of Integral, 637
Tags: trigonometry , calculus , integration , calculus computations , logarithm
For a non negative integer $n$, set t $I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx$ to answer the following questions: (1) Calculate $I_{n+2}+I_n.$ (2) Evaluate the values of $I_1,\ I_2$ and $I_3.$ 1978 Niigata university entrance exam

2014 PUMaC Number Theory A, 7
Tags: greatest common divisor , integration , number theory , floor function , modular arithmetic , calculus
Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.

2008 SEEMOUS, Problem 4
Tags: calculus , integration , inequalities
Let $n$ be a positive integer and $f:[0,1]\to\mathbb R$ be a continuous function such that $$\int^1_0x^kf(x)dx=1$$for every $k\in\{0,1,\ldots,n-1\}$. Prove that $$\int^1_0f(x)^2dx\ge n^2.$$

2011 Today's Calculation Of Integral, 742
Tags: calculus , integration , calculus computations
Evaluate \[\int_0^1 \frac{1-x^2}{(1+x^2)\sqrt{1+x^4}}\ dx\]

2005 Today's Calculation Of Integral, 60
Tags: trigonometry , calculus , integration , calculus computations
Let $a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)$ Evaluate \[\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})\]

2010 Today's Calculation Of Integral, 584
Tags: limit , calculus , calculus computations , integration , logarithm
Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

2023 CMIMC Integration Bee, 12
Tags: integration , calculus
\[\lim_{n\to\infty} n^2 \int_0^1 x^n e^{-x}\log(x)\,\mathrm dx\] [i]Proposed by Connor Gordon and Vlad Oleksenko[/i]

2007 Today's Calculation Of Integral, 224
Tags: calculus , integration , trigonometry , calculus computations
Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

2000 Romania National Olympiad, 2
Tags: real analysis , integration , integral , calculus , derivative
For any partition $ P $ of $ [0,1] $ , consider the set $$ \mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f’\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} . $$ Prove that there exists a partition $ P_0 $ of $ [0,1] $ such that $$ g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g’(x)\big| >4\cdot \# P. $$ Here, $ \# D $ denotes the natural number $ d $ such that $ 0=x_0<x_1<\cdots <x_d=1 $ is a partition $ D $ of $ [0,1] . $

2011 Today's Calculation Of Integral, 680
Tags: integration , calculus , calculus computations
Let $a>0$. Evaluate $\int_0^a x^2\left(1-\frac{x}{a}\right)^adx$. [i]2011 Keio University entrance exam/Science and Technology[/i]

2023 CMIMC Integration Bee, 15
Tags: calculus , integration
\[\int_0^\infty \left(1-e^{-\pi/x^2}\right)^2\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2002 Iran Team Selection Test, 9
Tags: function , number theory , integration , calculus , inequalities
$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?