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

2007 Today's Calculation Of Integral, 226
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {x^2}{(\cos x \plus{} x\sin x)^2}\ dx$ [color=darkblue]Virgil Nicula have already posted the integral[/color] :oops:

2010 Today's Calculation Of Integral, 582
Tags: calculus , integration , trigonometry , inequalities , calculus computations
Prove the following inequality. \[ \frac{\pi}{4}\sqrt{\frac{3}{2}\plus{}\sqrt{2}}<\int_0^{\frac{\pi}{2}} \sqrt{1\minus{}\frac 12\sin ^ 2 x}\ dx<\frac{\sqrt{3}}{4}\pi\]

2001 Federal Math Competition of S&M, Problem 1
Tags: calculus , derivative , number theory
Solve in positive integers \[ x^y + y = y^x + x \]

2010 Today's Calculation Of Integral, 580
Tags: calculus , integration , analytic geometry , geometry , calculus computations
Let $ k$ be a positive constant number. Denote $ \alpha ,\ \beta \ (0<\beta <\alpha)$ the $ x$ coordinates of the curve $ C: y\equal{}kx^2\ (x\geq 0)$ and two lines $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$.　Find the minimum area of the part bounded by the curve $ C$ and two lines $ l,\ m$.

1957 AMC 12/AHSME, 10
Tags: parabola , analytic geometry , quadratic , function , calculus , derivative , conic
The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its: $ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad \textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\ \textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad \textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\ \textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$

Today's calculation of integrals, 766
Tags: calculus , integration , function , trigonometry , inequalities , calculus computations
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2005 Today's Calculation Of Integral, 45
Tags: calculus , integration , function , trigonometry , calculus computations
Find the function $f(x)$ which satisfies the following integral equation. \[f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt\]

2010 Today's Calculation Of Integral, 621
Tags: calculus , integration , limit , real analysis , calculus computations , logarithm
Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$ [i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]

2006 Bundeswettbewerb Mathematik, 2
Tags: calculus , integration , geometry , 3d geometry
Prove that there are no integers $x,y$ for that it is $x^3+y^3=4\cdot(x^2y+xy^2+1)$.

2013 USA TSTST, 7
Tags: linear algebra , matrix , rotation , geometry , geometric transformation , vector , calculus
A country has $n$ cities, labelled $1,2,3,\dots,n$. It wants to build exactly $n-1$ roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly $1$, and it is also not permitted to put a road between cities $1$ and $n$. Let $T_n$ be the total number of possible ways to build these roads. (a) For all odd $n$, prove that $T_n$ is divisible by $n$. (b) For all even $n$, prove that $T_n$ is divisible by $n/2$.

2009 Today's Calculation Of Integral, 418
Tags: calculus , integration , calculus computations
(1) 2009 Kansai University entrance exam Calculate $ \int \frac{e^{\minus{}2x}}{1\plus{}e^{\minus{}x}}\ dx$. (2) 2009 Rikkyo University entrance exam/Science Evaluate $ \int_0^ 1 \frac{2x^3}{1\plus{}x^2}\ dx$.

2001 China Western Mathematical Olympiad, 1
Tags: floor function , function , calculus , quadratic , algebra
Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.

2022 CMIMC Integration Bee, 6
Tags: calculus , integration
\[\int_0^{2022} \{x\lfloor x \rfloor\}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2022 Romania National Olympiad, P3
Tags: calculus , function , functional inequality
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which are differentiable in $0$ and satisfy the following inequality for all real numbers $x,y$ \[f(x+y)+f(xy)\geq f(x)+f(y).\][i]Dan Ștefan Marinescu and Mihai Piticari[/i]

2003 Costa Rica - Final Round, 3
Tags: inequalities , induction , function , calculus
If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.

2012 Today's Calculation Of Integral, 804
Tags: calculus , integration , derivative , function , calculus computations , logarithm
For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$

2009 Today's Calculation Of Integral, 437
Tags: calculus , integration , calculus computations
Evaluate $ \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.$

2010 Today's Calculation Of Integral, 527
Tags: calculus , integration , calculus computations
Let $ n,\ m$ be positive integers and $ \alpha ,\ \beta$ be real numbers. Prove the following equations. (1) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)(x \minus{} \beta)\ dx \equal{} \minus{} \frac 16 (\beta \minus{} \alpha)^3$ (2) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)\ dx \equal{} \minus{} \frac {n!}{(n \plus{} 2)!}(\beta \minus{} \alpha)^{n \plus{} 2}$ (3) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)^mdx \equal{} ( \minus{} 1)^{m}\frac {n!m!}{(n \plus{} m \plus{} 1)!}(\beta \minus{} \alpha)^{n \plus{} m \plus{} 1}$

2020 Jozsef Wildt International Math Competition, W32
Tags: integration , calculus
Compute the quadruple integral $$A=\frac1{\pi^2}\int_{[0,1]^2\times[-\pi,\pi]^2}ab\sqrt{a^2+b^2-2ab\cos(x-y)}dadbdxdy$$ [i]Proposed by Moubinool Omarjee[/i]

2007 F = Ma, 33
Tags: calculus
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$. Find the maximum value of $\beta$. $ \textbf{(A)}\ 1$ $ \textbf{(B)}\ \sqrt{2}$ $ \textbf{(C)}\ 1/\sqrt{2}$ $ \textbf{(D)}\ \beta \text{ does not attain a maximum value}$ $ \textbf{(E)}\ \text{none of the above}$

2002 Tournament Of Towns, 1
Tags: trigonometry , calculus , integration , geometry
In a triangle $ABC$ it is given $\tan A,\tan B,\tan C$ are integers. Find their values.

2021 Alibaba Global Math Competition, 15
Tags: calculus , integration , topology , analysis , manifold , differential geometry
Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

1999 Harvard-MIT Mathematics Tournament, 5
Tags: calculus , quadratic , algebra , quadratic formula
Let $f(x)=x+\cfrac{1}{2x+\cfrac{1}{2x+\cfrac{1}{2x+\cdots}}}$. Find $f(99)f^\prime (99)$.

1997 Romania National Olympiad, 2
Tags: calculus , integration , function , real analysis
Prove that: $\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$ Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D

2011 China Girls Math Olympiad, 3
Tags: inequalities , function , calculus , hyperbola , conic
The positive reals $a,b,c,d$ satisfy $abcd=1$. Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}$.