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.

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Found problems: 2215

2007 F = Ma, 6
Tags: integration , calculus
At time $t = 0$ a drag racer starts from rest at the origin and moves along a straight line with velocity given by $v = 5t^2$, where $v$ is in $\text{m/s}$ and $t$ in $\text{s}$. The expression for the displacement of the car from $t = 0$ to time $t$ is $ \textbf{(A)}\ 5t^3 \qquad\textbf{(B)}\ 5t^3/3\qquad\textbf{(C)}\ 10t \qquad\textbf{(D)}\ 15t^2 \qquad\textbf{(E)}\ 5t/2 $

2006 Stanford Mathematics Tournament, 2
Tags: calculus , derivative , function
Find the minimum value of $ 2x^2\plus{}2y^2\plus{}5z^2\minus{}2xy\minus{}4yz\minus{}4x\minus{}2z\plus{}15$ for real numbers $ x$, $ y$, $ z$.

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

2010 Today's Calculation Of Integral, 523
Tags: logarithm , integration , calculus computations , calculus
Prove the following inequality. \[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]

2010 Romania National Olympiad, 1
Tags: limit , calculus , derivative , real analysis , function , integration
Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic function and $F:\mathbb{R}\to\mathbb{R}$ given by \[F(x)=\int_0^xf(t)\ \text{d}t.\] Prove that if $F$ has a finite derivative, then $f$ is continuous. [i]Dorin Andrica & Mihai Piticari[/i]

2019 Bangladesh Mathematical Olympiad, 6
Tags: calculus , number theory
When a function $f(x)$ is differentiated $n$ times ,the function we get id denoted $f^n(x)$.If $f(x)=\dfrac {e^x}{x}$.Find the value of \[\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}\]

2012 Today's Calculation Of Integral, 811
Tags: trigonometry , calculus , calculus computations , integration
Let $a$ be real number. Evaluate $\int_a^{a+\pi} |x|\cos x\ dx.$

2009 Today's Calculation Of Integral, 409
Tags: calculus computations , calculus , integration
Evaluate $ \int_0^1 \sqrt{\frac{x\plus{}\sqrt{x^2\plus{}1}}{x^2\plus{}1}}\ dx$.

1999 USAMTS Problems, 4
Tags: geometry , perimeter , calculus , analytic geometry , integration
We say a triangle in the coordinate plane is [i]integral[/i] if its three vertices have integer coordinates and if its three sides have integer lengths. (a) Find an integral triangle with perimeter of $42$. (b) Is there an integral triangle with perimeter of $43$?

2020 Jozsef Wildt International Math Competition, W6
Tags: calculus , integration
Determine the functions $f:(0,\pi)\to\mathbb R$ which satisfy $$f'(x)=\frac{\cos2020x}{\sin x}$$ for any real $x\in(0,\pi)$. [i]Proposed by D.M. Bătinețu-Giurgiu and Neculai Stanciu[/i]

1976 IMO Longlists, 14
Tags: number theory , integration , calculus
A sequence $\{ u_n \}$ of integers is defined by \[u_1 = 2, u_2 = u_3 = 7,\] \[u_{n+1} = u_nu_{n-1} - u_{n-2}, \text{ for }n \geq 3\] Prove that for each $n \geq 1$, $u_n$ differs by $2$ from an integral square.

2023 CMIMC Integration Bee, 9
Tags: calculus , integration
\[\int_{-1}^1 x^{2022}\cos\left(\tfrac \pi {12}-x\right)\sin\left(\tfrac \pi{12}+x\right)\,\mathrm dx\] [i]Proposed by Michael Duncan, Connor Gordon, and Vlad Oleksenko[/i]

2008 Grigore Moisil Intercounty, 3
Tags: inequalities , integration , calculus , function
Let $ f[0,\infty )\longrightarrow\mathbb{R} $ be a convex and differentiable function with $ f(0)=0. $ [b]a)[/b] Prove that $ \int_0^x f(t)dt\le \frac{x^2}{2}f'(x) , $ for any nonnegative $ x. $ [b]b)[/b] Determine $ f $ if the above inequality is actually an equality. [i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]

2010 Today's Calculation Of Integral, 646
Tags: logarithm , integration , calculus , calculus computations , trigonometry
Evaluate \[\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})\] Own

2013 Stanford Mathematics Tournament, 7
Tags: function , derivative , calculus , integration
The function $f(x)$ has the property that, for some real positive constant $C$, the expression \[\frac{f^{(n)}(x)}{n+x+C}\] is independent of $n$ for all nonnegative integers $n$, provided that $n+x+C\neq 0$. Given that $f'(0)=1$ and $\int_{0}^{1}f(x) \, dx = C+(e-2)$, determine the value of $C$. Note: $f^{(n)}(x)$ is the $n$-th derivative of $f(x)$, and $f^{(0)}(x)$ is defined to be $f(x)$.

2012 ISI Entrance Examination, 6
Tags: inequalities , conic , ellipse , calculus , perimeter , perpendicular bisector , geometry
[b]i)[/b] Let $0<a<b$.Prove that amongst all triangles having base $a$ and perimeter $a+b$ the triangle having two sides(other than the base) equal to $\frac {b}{2}$ has the maximum area. [b]ii)[/b]Using $i)$ or otherwise, prove that amongst all quadrilateral having give perimeter the square has the maximum area.

1995 Putnam, 2
Tags: calculus , integration
For what pairs of positive real numbers $(a,b)$ does the improper integral $(1)$ converge? \begin{align}\int_{b}^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)\,\mathrm{d}x \end{align}

2012 Today's Calculation Of Integral, 830
Tags: calculus computations , calculus , logarithm , limit , function , integration
Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

2011 Today's Calculation Of Integral, 744
Tags: calculus computations , inequalities , calculus , integration
Let $a,\ b$ be real numbers. If $\int_0^3 (ax-b)^2dx\leq 3$ holds, then find the values of $a,\ b$ such that $\int_0^3 (x-3)(ax-b)dx$ is minimized.

1981 Putnam, B6
Tags: double integral , calculus , integration
Let $C$ be a fixed unit circle in the cartesian plane. For any convex polygon $P$ , each of whose sides is tangent to $C$, let $N( P, h, k)$ be the number of points common to $P$ and the unit circle with center at $(h, k).$ Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the area of $H(P).$ Find the smallest number $u$ with $$ \frac{1}{F(P)} \int \int N(P,x,y)\;dx \;dy <u$$ for all polygons $P$, where the double integral is taken over $H(P).$

1994 AIME Problems, 10
Tags: trigonometry , ratio , inequalities , calculus , integration
In triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D.$ The lengths of the sides of $\triangle ABC$ are integers, $BD=29^3,$ and $\cos B=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2005 Today's Calculation Of Integral, 19
Tags: logarithm , calculus , trigonometry , calculus computations , integration
Calculate the following indefinite integrals. [1] $\int \tan ^ 3 x dx$ [2] $\int a^{mx+n}dx\ (a>0,a\neq 1, mn\neq 0)$ [3] $\int \cos ^ 5 x dx$ [4] $\int \sin ^ 2 x\cos ^ 3 x dx$ [5]$ \int \frac{dx}{\sin x}$

2010 Today's Calculation Of Integral, 621
Tags: calculus , logarithm , limit , real analysis , calculus computations , integration
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]

2009 Today's Calculation Of Integral, 413
Tags: integration , calculus computations , trigonometry , calculus , derivative
Find the maximum and minimum value of $ F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt$ for $ 0\leq x\leq \pi$.

2006 Cezar Ivănescu, 1
Tags: limit , factorial , harmonic series , newton s binom , integral , calculus
[b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^2}\sum_{i=0}^n\sqrt{\binom{n+i}{2}} $ [b]b)[/b] $ \lim_{n\to\infty } \frac{a^{H_n}}{1+n} ,\quad a>0 $