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: 2215

2013 Today's Calculation Of Integral, 861
Tags: analytic geometry , integration , function , geometry , calculus , calculus computations
Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

1951 AMC 12/AHSME, 15
Tags: integration , calculus
The largest number by which the expression $ n^3 \minus{} n$ is divisible for all possible integral values of $ n$, is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

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]

2021 JHMT HS, 10
Tags: derivative , polynomial , algebra , calculus
A polynomial $P(x)$ of some degree $d$ satisfies $P(n) = n^3 + 10n^2 - 12$ and $P'(n) = 3n^2 + 20n - 1$ for $n = -2, -1, 0, 1, 2.$ Also, $P$ has $d$ distinct (not necessarily real) roots $r_1, r_2, \ldots, r_d.$ The value of \[ \sum_{k=1}^{d}\frac{1}{4 - r_k^2} \] can be expressed as a common fraction $\tfrac{p}{q}.$ What is the value of $p + q?$

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 $\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$

2004 District Olympiad, 4
Tags: matrix , algebra , polynomial , calculus , derivative , linear algebra
Let $A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})$ such that $a_{12}=a_{23}=\ldots=a_{p-1,p}=1$ and $a_{ij}=0$ for any other entry. a)Prove that $A^{p-1}\neq O_p$ and $A^p=O_p$. b)If $X\in \mathcal{M}_{p}(\mathbb{C})$ and $AX=XA$, prove that there exist $a_1,a_2,\ldots,a_p\in \mathbb{C}$ such that: \[X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)\] c)If there exist $B,C\in \mathcal{M}_p(\mathbb{C})$ such that $(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*$, prove that $B=O_p$ or $C=O_p$.

1991 Baltic Way, 8
Tags: calculus , derivative , algebra , polynomial
Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation \[(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)\] \[+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)\] \[+(x - b)(x - c)(x - d)(x - e) = 0\] has four distinct real solutions.

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]

1970 IMO Longlists, 47
Tags: derivative , algebra , calculus , polynomial
Given a polynomial \[P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),\] where $a, b, c \neq 0$, prove that $P(x)$ is divisible by \[Q(x) = abx^2 + (a^2 + b^2)x + ab\] and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$.

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.

1994 Vietnam National Olympiad, 3
Tags: calculus , trigonometry , calculus computations
Define the sequence $\{x_{n}\}$ by $x_{0}=a\in (0,1)$ and $x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...)$. Show that the sequence converges and find its limit.

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$.

2019 District Olympiad, 1
Tags: calculus , limit , convergence , sequence
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n.$$

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.

2023 IMC, 1
Tags: derivative , calculus , function , functional equation
Find all functions $f: \mathbb{R} \to \mathbb{R}$ that have a continuous second derivative and for which the equality $f(7x+1)=49f(x)$ holds for all $x \in \mathbb{R}$.

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 \[\int_0^1 \frac{1-x^2}{(1+x^2)\sqrt{1+x^4}}\ dx\]

2018 PUMaC Live Round, Calculus 3
Tags: calculus
Let $\mathcal{R}(f(x))$ denote the number of distinct real roots of $f(x)$. Compute $$\sum_{a=1}^{1009}\sum_{b=1010}^{2018}\mathcal{R}(x^{2018}-ax^{2016}+b).$$