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

Tags were heavily modified to better represent problems.

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Found problems: 1687

2010 China Girls Math Olympiad, 8
Tags: calculus , integration , inequalities , modular arithmetic , number theory
Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$

1970 IMO Shortlist, 10
Tags: limit , inequality , calculus , integration , area
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

2011 Today's Calculation Of Integral, 738
Tags: calculus , integration , trigonometry , calculus computations
Answer the following questions: (1) Find the value of $a$ for which $S=\int_{-\pi}^{\pi} (x-a\sin 3x)^2dx$ is minimized, then find the minimum value. (2) Find the vlues of $p,\ q$ for which $T=\int_{-\pi}^{\pi} (\sin 3x-px-qx^2)^2dx$ is minimized, then find the minimum value.

2005 Today's Calculation Of Integral, 89
Tags: calculus , integration , trigonometry , function , symmetry , calculus computations
For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$ Find the value of $\frac{I_1}{I_2}.$

2008 Alexandru Myller, 3
Tags: integral , function , real analysis , inequalities , calculus , integration
Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

2009 VTRMC, Problem 3
Tags: calculus , integration , function
Define $f(x)=\int^x_0\int^x_0e^{u^2v^2}dudv$. Calculate $2f''(2)+f'(2)$.

2013 Today's Calculation Of Integral, 891
Tags: calculus , integration , geometry , 3d geometry , hyperbola , rotation , conic
Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

2009 IMS, 5
Tags: calculus , integration , function , real analysis
Suppose that $ f: \mathbb R^2\rightarrow \mathbb R$ is a non-negative and continuous function that $ \iint_{\mathbb R^2}f(x,y)dxdy\equal{}1$. Prove that there is a closed disc $ D$ with the least radius possible such that $ \iint_D f(x,y)dxdy\equal{}\frac12$.

1999 IberoAmerican, 1
Tags: calculus , integration , induction , number theory
Let $B$ be an integer greater than 10 such that everyone of its digits belongs to the set $\{1,3,7,9\}$. Show that $B$ has a [b]prime divisor[/b] greater than or equal to 11.

1962 AMC 12/AHSME, 22
Tags: calculus , integration
The number $ 121_b$, written in the integral base $ b$, is the square of an integer, for $ \textbf{(A)}\ b \equal{} 10,\text{ only} \qquad \textbf{(B)}\ b \equal{} 10 \text{ and } b \equal{} 5, \text{ only} \qquad \textbf{(C)}\ 2 \leq b \leq 10 \qquad \textbf{(D)}\ b > 2 \qquad \textbf{(E)}\ \text{no value of }b$

2010 Today's Calculation Of Integral, 636
Tags: calculus , integration , geometry , limit , derivative , calculus computations , logarithm
Let $a>1$ be a constant. In the $xy$-plane, let $A(a,\ 0),\ B(a,\ \ln a)$ and $C$ be the intersection point of the curve $y=\ln x$ and the $x$-axis. Denote by $S_1$ the area of the part bounded by the $x$-axis, the segment $BA$ and the curve $y=\ln x$ (1) For $1\leq b\leq a$, let $D(b,\ \ln b)$. Find the value of $b$ such that the area of quadrilateral $ABDC$ is the closest to $S_1$ and find the area $S_2$. (2) Find $\lim_{a\rightarrow \infty} \frac{S_2}{S_1}$. [i]1992 Tokyo University entrance exam/Science[/i]

2007 Today's Calculation Of Integral, 218
Tags: calculus , integration , quadratic , function , 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, 648
Tags: calculus , integration , function , calculus computations
Consider a function real-valued function with $C^{\infty}$-class on $\mathbb{R}$ such that: (a) $f(0)=\frac{df}{dx}(0)=0,\ \frac{d^2f}{dx^2}(0)\neq 0.$ (b) For $x\neq 0,\ f(x)>0.$ Judge whether the following integrals $(i),\ (ii)$ converge or diverge, justify your answer. $(i)$ \[\int\int_{|x_1|^2+|x_2|^2\leq 1} \frac{dx_1dx_2}{f(x_1)+f(x_2)}.\] $(ii)$ \[\int\int_{|x_1|^2+|x_2|^2+|x_3|^2\leq 1} \frac{dx_1dx_2dx_3}{f(x_1)+f(x_2)+f(x_3)}.\] [i]2010 Kyoto University, Master Course in Mathematics[/i]

1999 Putnam, 5
Tags: polynomial , integration , calculus , inequalities , real analysis
Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]

2013 Today's Calculation Of Integral, 884
Tags: calculus , integration , trigonometry , inequalities , calculus computations , integral inequality
Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

2004 Harvard-MIT Mathematics Tournament, 10
Tags: calculus , integration , trigonometry , induction
Let $P(x)=x^3-\tfrac{3}{2}x^2+x+\tfrac{1}{4}$. Let $P^{[1]}(x)=P(x)$, and for $n\ge1$, let $P^{n+1}(x)=P^{[n]}(P(x))$. Evaluate: \[ \displaystyle\int_{0}^{1} P^{[2004]} (x) \ \mathrm{d}x. \]

2011 India National Olympiad, 3
Tags: algebra , polynomial , calculus , integration , rational root theorem , number theory
Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0$ be two polynomials with integral coefficients such that $a_n-b_n$ is a prime and $a_nb_0-a_0b_n\neq 0,$ and $a_{n-1}=b_{n-1}.$ Suppose that there exists a rational number $r$ such that $P(r)=Q(r)=0.$ Prove that $r\in\mathbb Z.$

2023 CMIMC Integration Bee, 2
Tags: calculus , integration
\[\int_0^1 \frac{1}{x+\sqrt x}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2013 Today's Calculation Of Integral, 897
Tags: calculus , integration , geometry , geometric transformation , rotation , calculus computations , logarithm
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

2013 Today's Calculation Of Integral, 890
Tags: calculus , integration , function , algorithm , calculus computations
A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

Today's calculation of integrals, 897
Tags: calculus , integration , geometry , geometric transformation , rotation , calculus computations , logarithm
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

2009 Unirea, 4
Tags: limit , trigonometry , geometry , modular arithmetic , probability , integration , logarithm
Evaluate the limit: \[ \lim_{n \to \infty}{n \cdot \sin{1} \cdot \sin{2} \cdot \dots \cdot \sin{n}}.\] Proposed to "Unirea" Intercounty contest, grade 11, Romania

2007 Princeton University Math Competition, 2
Tags: calculus , integration
Find the biggest non-integer $x$ such that $(x+2)^2 + (x+3)^3 + (x+4)^4 = 2$.

2009 Today's Calculation Of Integral, 510
Tags: calculus , integration , trigonometry , function , derivative , calculus computations
(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$. (2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.

1986 National High School Mathematics League, 3
Tags: calculus , integration
In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it integral point. Please color all intengral points in white, red and black, satisfying: (1) Points in every color appear on infinitely many lines that are parallel to $x$-axis. (2) For any white point $A$, red point $B$, black point $C$, we can find another red point $D$, such that $ABCD$ is a parallelogram.