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

2010 Romania National Olympiad, 1
Tags: calculus , derivative , function , integration , limit , real analysis
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 District Olympiad, 2
Tags: integrability , function , calculus
Let $n$ be a positive integer and $f:[0,1] \to \mathbb{R}$ be an integrable function. Prove that there exists a point $c \in \left[0,1- \frac{1}{n} \right],$ such that [center] $ \int\limits_c^{c+\frac{1}{n}}f(x)\mathrm{d}x=0$ or $\int\limits_0^c f(x) \mathrm{d}x=\int\limits_{c+\frac{1}{n}}^1f(x)\mathrm{d}x.$ [/center]

2015 BMT Spring, 15
Tags: calculus , integration
Compute $$\int_{1/2}^{2} \frac{x^2 + 1}{x^2(x^{2015} + 1)} dx.$$

2010 Today's Calculation Of Integral, 610
Tags: calculus , integration , calculus computations , logarithm
Evaluate $\int_2^a \frac{x^a-1-xa^x\ln a}{(x^a-1)^2}dx.$ proposed by kunny

2018 VTRMC, 5
Tags: calculus , integration , sequence
For $n \in \mathbb{N}$, let $a_n = \int _0 ^{1/\sqrt{n}} | 1 + e^{it} + e^{2it} + \dots + e^{nit} | \ dt$. Determine whether the sequence $(a_n) = a_1, a_2, \dots$ is bounded.

2012 Today's Calculation Of Integral, 849
Tags: calculus , integration , function , calculus computations
Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

2021 Simon Marais Mathematical Competition, A4
Tags: calculus , real analysis
For each positive real number $r$, define $a_0(r) = 1$ and $a_{n+1}(r) = \lfloor ra_n(r) \rfloor$ for all integers $n \ge 0$. (a) Prove that for each positive real number $r$, the limit \[ L(r) = \lim_{n \to \infty} \frac{a_n(r)}{r^n} \] exists. (b) Determine all possible values of $L(r)$ as $r$ varies over the set of positive real numbers. [i]Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.[/i]

1997 VJIMC, Problem 3
Tags: calculus , integration , inequalities , multivariable calculus
Let $u\in C^2(\overline D)$, $u=0$ on $\partial D$ where $D$ is the open unit ball in $\mathbb R^3$. Prove that the following inequality holds for all $\varepsilon>0$: $$\int_D|\nabla u|^2dV\le\varepsilon\int_D(\Delta u)^2dV+\frac1{4\varepsilon}\int_Du^2dV.$$(We recall that $\nabla u$ and $\Delta u$ are the gradient and Laplacian, respectively.)

PEN Q Problems, 6
Tags: binomial coefficient , polynomial , calculus , derivative , binomial theorem , algebra
Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.

2004 Harvard-MIT Mathematics Tournament, 1
Tags: calculus , trigonometry , limit
Let $f(x)=\sin(\sin(x))$. Evaluate \[ \lim_{h \to 0} \dfrac {f(x+h)-f(h)}{x} \] at $x=\pi$.

2006 Harvard-MIT Mathematics Tournament, 8
Tags: calculus , integration , trigonometry , logarithm
Compute $\displaystyle\int_0^{\pi/3}x\tan^2(x)dx$.

2006 Petru Moroșan-Trident, 3
Tags: indefinite integral , calculus , primitive
Determine the primitives of: [b]1)[/b] $ (0,\pi /2)\ni x\mapsto\frac{x^2}{-x+\tan x} $ [b]2)[/b] $ 1<x\mapsto \frac{-1+\ln x}{x^2-\ln^2 x} $ [i]Ion Nedelcu[/i]

2012 ELMO Shortlist, 6
Tags: inequalities , induction , calculus
Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$. [i]Calvin Deng.[/i]

1940 Putnam, A2
Tags: calculus , derivative
Let $A,B$ be two fixed points on the curve $y=f(x)$, $f$ is continuous with continuous derivative and the arc $\widehat{AB}$ is concave to the chord $AB$. If $P$ is a point on the arc $\widehat{AB}$ for which $AP+PB$ is maximal, prove that $PA$ and $PB$ are equally inclined to the tangent to the curve $y=f(x)$ at $P$.

2014 NIMO Problems, 8
Tags: analytic geometry , graphing lines , slope , calculus , derivative , hmmt , trigonometry
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$. [i]Proposed by Akshaj[/i]

1950 Miklós Schweitzer, 10
Tags: advanced fields , calculus , derivative
Consider an arc of a planar curve such that the total curvature of the arc is less than $ \pi$. Suppose, further, that the curvature and its derivative with respect to the arc length exist at every point of the arc and the latter nowhere equals zero. Let the osculating circles belonging to the endpoints of the arc and one of these points be given. Determine the possible positions of the other endpoint.

1975 AMC 12/AHSME, 22
Tags: calculus , integration , quadratic
If $ p$ and $ q$ are primes and $ x^2 \minus{} px \plus{} q \equal{} 0$ has distinct positive integral roots, then which of the following statements are true? $ \text{I. The difference of the roots is odd.}$ $ \text{II. At least one root is prime.}$ $ \text{III. } p^2 \minus{} q \text{ is prime.}$ $ \text{IV. } p \plus{} q \text{ is prime.}$ $ \textbf{(A)}\ \text{I only} \qquad \textbf{(B)}\ \text{II only} \qquad \textbf{(C)}\ \text{II and III only} \qquad$ $ \textbf{(D)}\ \text{I, II and IV only} \qquad \textbf{(E)}\ \text{All are true.}$

1960 AMC 12/AHSME, 11
Tags: calculus , integration , vieta
For a given value of $k$ the product of the roots of \[ x^2-3kx+2k^2-1=0 \] is $7$. The roots may be characterized as: $ \textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad$ $\textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary} $

2004 District Olympiad, 4
Tags: integral , real analysis , calculus , integration
Let $ a,b\in (0,1) $ and a continuous function $ f:[0,1]\longrightarrow\mathbb{R} $ with the property that $$ \int_0^x f(t)dt=\int_0^{ax} f(t)dt +\int_0^{bx} f(t)dt,\quad\forall x\in [0,1] . $$ [b]a)[/b] Show that if $ a+b<1, $ then $ f=0. $ [b]b)[/b] Show that if $ a+b=1, $ then $ f $ is constant.

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

2021 CMIMC Integration Bee, 9
Tags: calculus , integration
$$\int_1^2\frac{12x^3+12x+12}{2x^4+3x^2+4x}\,dx$$ [i]Proposed by Connor Gordon[/i]

2009 Today's Calculation Of Integral, 517
Tags: calculus , integration , geometry , search , calculus computations
Consider points $ P$ which are inside the square with side length $ a$ such that the distance from $ P$ to the center of the square equals to the least distance from $ P$ to each side of the square.Find the area of the figure formed by the whole points $ P$.

2003 China Team Selection Test, 1
Tags: trigonometry , calculus , geometry
Let $ ABCD$ be a quadrilateral which has an incircle centered at $ O$. Prove that \[ OA\cdot OC\plus{}OB\cdot OD\equal{}\sqrt{AB\cdot BC\cdot CD\cdot DA}\]

2013 SEEMOUS, Problem 1
Tags: function , calculus , integration
Find all continuous functions $f:[1,8]\to\mathbb R$, such that $$\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.$$

2009 Today's Calculation Of Integral, 404
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_{ \minus{} \pi}^{\pi} \frac {\sin nx}{(1 \plus{} 2009^x)\sin x}\ dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.