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

2010 VJIMC, Problem 3
Tags: calculus , integration
Prove that there exist positive constants $c_1$ and $c_2$ with the following properties: a) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\cos(kx)\text dx\right|<\frac{c_1}{k^{3/2}}.$$b) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\sin(kx)\text dx\right|<\frac{c_2}k.$$

2022 JHMT HS, 7
Tags: calculus , integration
Let $a$ be the unique real number $x$ satisfying $xe^x = 2$. Find a closed-form expression for \[ \int_{a}^{\infty} \frac{x + 1}{x\sqrt{(xe^x)^{11} - 1}}\,dx. \] You may express your answer in terms of elementary operations, functions, and constants.

2007 Estonia National Olympiad, 3
Tags: calculus , integration , analytic geometry , geometry
Does there exist an equilateral triangle (a) on a plane; (b) in a 3-dimensional space; such that all its three vertices have integral coordinates?

2004 China Team Selection Test, 3
Tags: algebra , polynomial , calculus , integration , number theory
Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

2009 Today's Calculation Of Integral, 458
Tags: calculus , integration , geometry , vector , calculus computations , logarithm
Let $ S(t)$ be the area of the traingle $ OAB$ with $ O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t)$. Evaluate $ \int_1^ e S(t)^2\ln t\ dt$.

2014 Putnam, 2
Tags: function , integration , real analysis , absolute value
Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

2010 Today's Calculation Of Integral, 628
Tags: calculus , integration , trigonometry , geometry , function , calculus computations
(1) Evaluate the following definite integrals. (a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$ (b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$ (c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$ (2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that \[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\] , then find the volume of the solid. [i]1984 Yamanashi Medical University entrance exam[/i] Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them. Thanks in advance. kunny

2007 Today's Calculation Of Integral, 193
Tags: calculus , integration , parabola , geometry , limit , analytic geometry , conic
For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$. Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin. (1) Find the equation of $l$. (2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.

2023 CMIMC Integration Bee, 1
Tags: calculus , integration
\[\int_2^0 x^2+3\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2013 Today's Calculation Of Integral, 861
Tags: calculus , integration , geometry , function , analytic geometry , calculus computations
Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

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

2022 VJIMC, 3
Tags: calculus , integration , function
Let $f:[0,1]\to\mathbb R$ be a given continuous function. Find the limit $$\lim_{n\to\infty}(n+1)\sum_{k=0}^n\int^1_0x^k(1-x)^{n-k}f(x)dx.$$

2018 Romania National Olympiad, 2
Tags: function , calculus , integration , real analysis
Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$ For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$ Determine $\min_{f \in \mathcal{F}}I(f).$ [i]Liviu Vlaicu[/i]

1989 IMO Longlists, 27
Tags: calculus , integration , modular arithmetic , number theory , relatively prime , combinatorics
Let $ L$ denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points $ A,B,C$ of $ L$ there is a fourth point $ D,$ different from $ A,B,C,$ such that the interiors of the segments $ AD,BD,CD$ contain no points of $ L.$ Is the statement true if one considers four points of $ L$ instead of three?

2007 Today's Calculation Of Integral, 248
Tags: calculus , integration , trigonometry , function , trig identities , calculus computations
Evaluate $ \int_{\frac {\pi}{4}}^{\frac {3}{4}\pi } \cos \frac {1}{\sin \left(\frac {1}{\sin x}\right)}\cdot \cos \left(\frac {1}{\sin x}\right)\cdot \frac {\cos x}{\sin ^ 2 x\cdot \sin ^ 2 \left(\frac {1}{\sin x }\right)}\ dx$ Last Edited, Sorry kunny

Today's calculation of integrals, 887
Tags: calculus , integration , trigonometry , derivative , calculus computations , logarithm , function
For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

1963 Putnam, A3
Tags: calculus , integration , function , differential equation
Find an integral formula for the solution of the differential equation $$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$ for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$

2010 IMC, 1
Tags: integration , real analysis
Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

2007 Today's Calculation Of Integral, 198
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Compare the values of the following definite integrals. \[\int_{0}^{\infty}\ln \left(x+\frac{1}{x}\right)\frac{dx}{1+x^{2}},\ \ \int_{0}^{\frac{\pi}{2}}\left(\frac{\theta}{\sin \theta}\right)^{2}d\theta\]

2014 Contests, 2
Tags: calculus , integration , parabola , geometry , calculus computations , conic
Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

1964 Putnam, A2
Tags: integration , system of equations
Find all continuous positive functions $f(x)$, for $0\leq x \leq 1$, such that $$\int_{0}^{1} f(x)\; dx =1, $$ $$\int_{0}^{1} xf(x)\; dx =\alpha,$$ $$\int_{0}^{1} x^2 f(x)\; dx =\alpha^2, $$ where $\alpha$ is a given real number.

2010 Today's Calculation Of Integral, 667
Tags: calculus , integration , geometric transformation , rotation , trigonometry , logarithm , geometry
Let $a>1,\ 0\leq x\leq \frac{\pi}{4}$. Find the volume of the solid generated by a rotation of the part bounded by two curves $y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}$ about the $x$-axis. [i]1993 Hiroshima Un iversity entrance exam/Science[/i]

2005 Today's Calculation Of Integral, 53
Tags: calculus , integration , trigonometry , calculus computations
Find the maximum value of the following integral. \[\int_0^{\infty} e^{-x}\sin tx\ dx\]

2010 Today's Calculation Of Integral, 659
Tags: calculus , integration , geometric series , calculus computations , logarithm
Evaluate $\int_0^1 \frac{\ln (x+2)}{x+1}dx.$

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