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

2013 Today's Calculation Of Integral, 883
Tags: calculus , integration , trigonometry , calculus computations , integral inequality , logarithm
Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

1997 Traian Lălescu, 4
Tags: limit , trigonometry , calculus , integration , real analysis
Compute the limit: \[ \lim_{n\to\infty} \frac{1}{n^2}\sum\limits_{1\leq i <j\leq n}\sin \frac{i+j}{n}\].

Today's calculation of integrals, 899
Tags: calculus , integration , limit , calculus computations
Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2010 Today's Calculation Of Integral, 541
Tags: calculus , integration , function , calculus computations
Find the functions $ f(x),\ g(x)$ satisfying the following equations. (1) $ f'(x) \equal{} 2f(x) \plus{} 10,\ f(0) \equal{} 0$ (2) $ \int_0^x u^3g(u)du \equal{} x^4 \plus{} g(x)$

1995 AMC 12/AHSME, 23
Tags: inequalities , calculus , integration , trigonometry , triangle inequality , pythagorean theorem , geometry
The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

2009 Today's Calculation Of Integral, 505
Tags: calculus , integration , trigonometry , calculus computations
In the $ xyz$ space with the origin $ O$, given a cuboid $ K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2$ and the plane $ \alpha : z \equal{} 2$. Draw the perpendicular $ PH$ from $ P$ to the plane. Find the volume of the solid formed by all points of $ P$ which are included in $ K$ such that $ \overline{OP}\leq \overline{PH}$.

2013 Stanford Mathematics Tournament, 8
Tags: calculus , function , analytic geometry , graphing lines , slope , integration
The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, it intersects the graph $y=f(x)$ at precisely one point, which is $\frac{1}{\sqrt{m}}$ units from the origin. Let $a$ be the unique real number for which $f$ takes on its maximum value at $x=a$ (you may assume that such an $a$ exists). Find $\int_{0}^{a}f(x) \, dx$.

2012 Today's Calculation Of Integral, 779
Tags: calculus , integration , parabola , analytic geometry , geometry , calculus computations , conic
Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane. When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.

2011 Today's Calculation Of Integral, 721
Tags: calculus , integration , function , calculus computations
For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.

2010 ELMO Shortlist, 4
Tags: integration , combinatorics
The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$. [i]Brian Hamrick.[/i]

2007 Today's Calculation Of Integral, 249
Tags: calculus , integration , calculus computations , logarithm
Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.

2024 CMIMC Integration Bee, 5
Tags: calculus , integration
\[\int_1^e \frac{2x^2+1}{x^3+x\log(x)}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2011 Today's Calculation Of Integral, 717
Tags: calculus , integration , calculus computations , logarithm , geometry
Let $a_n$ be the area of the part enclosed by the curve $y=x^n\ (n\geq 1)$, the line $x=\frac 12$ and the $x$ axis. Prove that : \[0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}\]

2005 Today's Calculation Of Integral, 44
Tags: calculus , integration , trigonometry , induction , abstract algebra , calculus computations
Evaluate \[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]

2010 Today's Calculation Of Integral, 525
Tags: calculus , integration , geometry , vector , inequalities , quadratic , trigonometry
Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

2014 Contests, 903
Tags: calculus , integration , function , real analysis , geometric series , calculus computations
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$

2017 VJIMC, 4
Tags: calculus , integration , real analysis
Let $f:(1,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $f(x) \le x^2 \log(x)$ and $f'(x)>0$ for every $x \in (1,\infty)$. Prove that \[\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.\]

1977 Miklós Schweitzer, 9
Tags: vector , function , integration , real analysis
Suppose that the components of he vector $ \textbf{u}=(u_0,\ldots,u_n)$ are real functions defined on the closed interval $ [a,b]$ with the property that every nontrivial linear combination of them has at most $ n$ zeros in $ [a,b]$. Prove that if $ \sigma$ is an increasing function on $ [a,b]$ and the rank of the operator \[ A(f)= \int_{a}^b \textbf{u}(x)f(x)d\sigma(x), \;f \in C[a,b]\ ,\] is $ r \leq n$, then $ \sigma$ has exactly $ r$ points of increase. [i]E. Gesztelyi[/i]

2005 Today's Calculation Of Integral, 33
Tags: calculus , integration , trigonometry , function , derivative , calculus computations , logarithm
Evaluate \[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]

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

2010 Today's Calculation Of Integral, 658
Tags: calculus , integration , trigonometry , geometry , analytic geometry , calculus computations
Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$ (1) Find the length $L$ of $C$. (2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$. Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School. [i]1997 Kyoto University entrance exam/Science[/i]

2000 USA Team Selection Test, 4
Tags: induction , function , calculus , integration
Let $n$ be a positive integer. Prove that \[ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right). \]

Today's calculation of integrals, 886
Tags: calculus , integration , function , trigonometry , calculus computations
Find the functions $f(x),\ g(x)$ such that $f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$ $g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$

2008 Moldova MO 11-12, 2
Tags: integration , trigonometry , real analysis
Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.

2022 CMIMC Integration Bee, 8
Tags: calculus , integration
\[\int_{-\infty}^{0} \frac{1}{e^{-x}+2e^{x}+e^{3x}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]