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.

AND:
OR:
NO:

Found problems: 2215

2009 Today's Calculation Of Integral, 406
Tags: calculus , integration , limit , trigonometry , calculus computations
Find $ \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos (2n\plus{}1)x|\ dx$.

2012 IFYM, Sozopol, 4
Tags: algebra , polynomial , calculus , integration , gauss , number theory , prime number
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

2018 Romania National Olympiad, 2
Tags: calculus , inequalities
Let $x>0.$ Prove that $$2^{-x}+2^{-1/x} \leq 1.$$

2006 Iran MO (3rd Round), 3
Tags: abstract algebra , group theory , calculus , integration , invariant , number theory
$L$ is a fullrank lattice in $\mathbb R^{2}$ and $K$ is a sub-lattice of $L$, that $\frac{A(K)}{A(L)}=m$. If $m$ is the least number that for each $x\in L$, $mx$ is in $K$. Prove that there exists a basis $\{x_{1},x_{2}\}$ for $L$ that $\{x_{1},mx_{2}\}$ is a basis for $K$.

2012 Today's Calculation Of Integral, 844
Tags: calculus , integration , limit , calculus computations
Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find $\lim_{n\to\infty} n^2I_n.$

2010 Today's Calculation Of Integral, 593
Tags: calculus , integration , trigonometry , function , calculus computations
For a positive integer $m$, prove the following ineqaulity. $0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$ 1996 Osaka University entrance exam

2010 Today's Calculation Of Integral, 596
Tags: calculus , integration , trigonometry , calculus computations
Find the minimum value of $\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).$ 2009 Shimane University entrance exam/Medicine

2009 Today's Calculation Of Integral, 438
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_{\sqrt{2}\minus{}1}^{\sqrt{2}\plus{}1} \frac{x^4\plus{}x^2\plus{}2}{(x^2\plus{}1)^2}\ dx.$

2014 VTRMC, Problem 2
Tags: integration , calculus
Evaluate $\int^2_0\frac{x(16-x^2)}{16-x^2+\sqrt{(4-x)(4+x)(12+x^2)}}dx$.

2011 Today's Calculation Of Integral, 740
Tags: calculus , integration , geometry , derivative , calculus computations
Let $r$ be a positive constant. If 2 curves $C_1: y=\frac{2x^2}{x^2+1},\ C_2: y=\sqrt{r^2-x^2}$ have each tangent line at their point of intersection and at which their tangent lines are perpendicular each other, then find the area of the figure bounded by $C_1,\ C_2$.

2023 CMIMC Integration Bee, 8
Tags: calculus , integration
\[\int_{-10}^{10}|4-|3-|2-|1-|x|||||\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2010 Today's Calculation Of Integral, 529
Tags: calculus , integration , inequalities , trigonometry , calculus computations
Prove that the following inequality holds for each natural number $ n$. \[ \int_0^{\frac {\pi}{2}} \sum_{k \equal{} 1}^n \left(\frac {\sin kx}{k}\right)^2dx < \frac {61}{144}\pi\]

2011 Today's Calculation Of Integral, 707
Tags: calculus , integration , calculus computations
In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that \[\left\{ \begin{array}{ll} x^2+y^2\leq 4 &\quad \\ 0\leq z\leq 4 &\quad \end{array} \right.\] Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying \[\left\{ \begin{array}{ll} z\leq (x-2)^2 &\quad \\ z\leq y^2 &\quad \end{array} \right.\] Find the volume of the solid $V$.

2001 District Olympiad, 4
Tags: limit , integration , calculus , logarithm , real analysis
a)Prove that $\ln(1+x)\le x,\ (\forall)x\ge 0$. b)Let $a>0$. Prove that: \[\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}\] [i]***[/i]

2007 Today's Calculation Of Integral, 212
Tags: calculus , integration , trigonometry , calculus computations
For integers $k\ (0\leq k\leq 5)$, positive numbers $m,\ n$ and real numbers $a,\ b$, let $f(k)=\int_{-\pi}^{\pi}(\sin kx-a\sin mx-b\sin nx)^{2}\ dx$, $p(k)=\frac{5!}{k!(5-k)!}\left(\frac{1}{2}\right)^{5}, \ E=\sum_{k=0}^{5}p(k)f(k)$. Find the values of $m,\ n,\ a,\ b$ for which $E$ is minimized.

2019 Brazil Undergrad MO, 3
Tags: calculus , system of equations , algebra
Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations $2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}$ $x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}$ $x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}$ have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.

2005 Harvard-MIT Mathematics Tournament, 2
Tags: function , calculus , derivative
How many real numbers $x$ are solutions to the following equation? \[ 2003^x + 2004^x = 2005^x \]

2009 Today's Calculation Of Integral, 441
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_1^e \frac{(x^2\ln x\minus{}1)e^x}{x}\ dx.$

1997 AIME Problems, 7
Tags: analytic geometry , graphing lines , slope , calculus , conic
A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0,$ the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2).$

2007 Princeton University Math Competition, 8
Tags: geometry , calculus , integration , ellipse , ratio , conic
What is the area of the region defined by $x^2+3y^2 \le 4$ and $y^2+3x^2 \le 4$?

2007 Today's Calculation Of Integral, 170
Tags: calculus , integration , algebra , partial fractions , calculus computations , 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) $J=\int \frac{dx}{(x^{2}+2ax+b)^{2}}$

2007 Hungary-Israel Binational, 3
Tags: trigonometry , function , calculus , derivative , inequalities , geometry
Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.

2004 USAMTS Problems, 2
Tags: geometry , 3d geometry , calculus , integration
For the equation \[ (3x^2+y^2-4y-17)^3-(2x^2+2y^2-4y-6)^3=(x^2-y^2-11)^3, \] determine its solutions $(x, y)$ where both $x$ and $y$ are integers. Prove that your answer lists all the integer solutions.

2006 Putnam, B5
Tags: function , integration , calculus , quadratic , inequalities , algebra
For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$

2010 Today's Calculation Of Integral, 640
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $\int_0^{\frac{\pi}{4}} \frac{1}{1-\sin x}\sqrt{\frac{\cos x}{1+\cos x+\sin x}}dx.$ Own