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

2010 Today's Calculation Of Integral, 550
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}$.

2007 Today's Calculation Of Integral, 209
Tags: calculus , integration , trigonometry , calculus computations
Let $m,\ n$ be the given distinct positive integers. Answer the following questions. (1) Find the real number $\alpha \ (|\alpha |<1)$ such that $\int_{-\pi}^{\pi}\sin (m+\alpha )x\ \sin (n+\alpha )x\ dx=0$. (2) Find the real number $\beta$ satifying the sytem of equation $\int_{-\pi}^{\pi}\sin^{2}(m+\beta )x\ dx=\pi+\frac{2}{4m-1}$, $\int_{-\pi}^{\pi}\sin^{2}(n+\beta )x\ dx=\pi+\frac{2}{4n-1}$.

2020 Jozsef Wildt International Math Competition, W3
Tags: integration , real analysis , calculus
Let $n \geq 2$ be an integer. Calculate$$\int \limits_{0}^{\frac{\pi}{2}}\frac{\sin x}{\sin^{2n-1}x+\cos^{2n-1}x}dx$$

2007 Romania National Olympiad, 4
Tags: function , calculus , derivative , symmetry , ratio , real analysis
Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$. a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$. b) Give an example of a non-constant function $f$ with property $(P)$. c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

2010 Today's Calculation Of Integral, 644
Tags: calculus , integration , trigonometry , inequalities , calculus computations , logarithm
For a constant $p$ such that $\int_1^p e^xdx=1$, prove that \[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\] Own

2010 Today's Calculation Of Integral, 585
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.

2009 Vietnam Team Selection Test, 1
Tags: calculus , quadratic , function , logarithm , inequalities
Let $ a,b,c$ be positive numbers.Find $ k$ such that: $ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$

2012 Today's Calculation Of Integral, 816
Tags: calculus , integration , geometry , geometric transformation , rotation , combinatorial geometry , calculus computations
Find the volume of the solid of a circle $x^2+(y-1)^2=4$ generated by a rotation about the $x$-axis.

2011 Today's Calculation Of Integral, 720
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.

1967 Miklós Schweitzer, 9
Tags: calculus , derivative , symmetry , advanced fields
Let $ F$ be a surface of nonzero curvature that can be represented around one of its points $ P$ by a power series and is symmetric around the normal planes parallel to the principal directions at $ P$. Show that the derivative with respect to the arc length of the curvature of an arbitrary normal section at $ P$ vanishes at $ P$. Is it possible to replace the above symmetry condition by a weaker one? [i]A. Moor[/i]

1994 AIME Problems, 4
Tags: floor function , calculus , derivative , geometric series , logarithm
Find the positive integer $n$ for which \[ \lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994. \] (For real $x$, $\lfloor x\rfloor$ is the greatest integer $\le x.$)

2010 Today's Calculation Of Integral, 617
Tags: calculus , integration , function , calculus computations
Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane. Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ [i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]

2010 Today's Calculation Of Integral, 595
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$ 2009 Kumamoto University entrance exam/Medicine

1990 Turkey Team Selection Test, 2
Tags: algebra , polynomial , calculus , derivative , inequalities
For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!) For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.

1996 IMC, 5
Tags: function , calculus , derivative , real analysis
i) Let $a,b$ be real numbers such that $b\leq 0$ and $1+ax+bx^{2} \geq 0$ for every $x\in [0,1]$. Prove that $$\lim_{n\to \infty} n \int_{0}^{1}(1+ax+bx^{2})^{n}dx= \begin{cases} -\frac{1}{a} &\text{if}\; a<0,\\ \infty & \text{if}\; a \geq 0. \end{cases}$$ ii) Let $f:[0,1]\rightarrow[0,\infty)$ be a function with a continuous second derivative and let $f''(x)\leq0$ for every $x\in [0,1]$. Suppose that $L=\lim_{n\to \infty} n \int_{0}^{1}(f(x))^{n}dx$ exists and $0<L<\infty$. Prove that $f'$ has a constant sign and $\min_{x\in [0,1]}|f'(x)|=L^{-1}$.

2005 China National Olympiad, 4
Tags: inequalities , calculus , function , derivative , algebra , logarithm
The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]

2010 Victor Vâlcovici, 2
Tags: function , calculus , logarithm , inequalities
Let $ f:[2,\infty )\rightarrow\mathbb{R} $ be a differentiable function satisfying $ f(2)=0 $ and $$ \frac{df}{dx}=\frac{2}{x^2+f^4{x}} , $$ for any $ x\in [2,\infty ) . $ Show that there exists $ \lim_{x\to\infty } f(x) $ and is at most $ \ln 3. $ [i]Gabriel Daniilescu[/i]

1983 AIME Problems, 9
Tags: trigonometry , inequalities , calculus
Find the minimum value of \[\frac{9x^2 \sin^2 x + 4}{x \sin x}\] for $0 < x < \pi$.

2012 Iran Team Selection Test, 1
Tags: algebra , polynomial , modular arithmetic , induction , vieta , calculus
Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$-remainder if $ \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$. Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$-remainder and the polynomial $(f(x))^{p-1}$ is $1$-remainder. [i]Proposed by Yahya Motevassel[/i]

2011 Today's Calculation Of Integral, 678
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\int_0^{\pi} \left(1+\sum_{k=1}^n k\cos kx\right)^2dx\ \ (n=1,\ 2,\ \cdots).\] [i]2011 Doshisya University entrance exam/Life Medical Sciences[/i]

1985 AMC 12/AHSME, 21
Tags: calculus , integration
How many integers $ x$ satisfy the equation \[ (x^2 \minus{} x \minus{} 1)^{x \plus{} 2} \equal{} 1 \]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{none of these}$

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]

2019 Jozsef Wildt International Math Competition, W. 31
Tags: integration , inequalities , calculus
Let $a, b \in \Gamma$, $a < b$ and the differentiable function $f : [a, b] \to \Gamma$, such that $f (a) = a$ and $f (b) = b$. Prove that $$\int \limits_{a}^{b} \left(f'(x)\right)^2dx \geq b-a$$

2005 Today's Calculation Of Integral, 63
Tags: calculus , integration , limit , trigonometry , calculus computations
For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$ Evaluate \[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]

2013 ELMO Shortlist, 5
Tags: function , calculus , derivative , 3-variable inequality , logarithm , inequalities
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]