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

2011 Today's Calculation Of Integral, 768
Tags: calculus , integration , limit , geometry , 3d geometry , sphere , symmetry
Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying \[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\] in $xyz$-space. (1) Find $V(r)$. (2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$ (3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

2008 Harvard-MIT Mathematics Tournament, 2
Tags: analytic geometry , graphing lines , slope , calculus , derivative , calculus computations
([b]3[/b]) Let $ \ell$ be the line through $ (0,0)$ and tangent to the curve $ y \equal{} x^3 \plus{} x \plus{} 16$. Find the slope of $ \ell$.

2007 Romania National Olympiad, 1
Tags: calculus , integration , inequalities , function , derivative , trigonometry , logarithm
Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained. [hide="Comment"] In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''. [/hide]

2024-IMOC, A6
Tags: inequalities , calculus
Given positive real $a,b,c$ satisfying \[\frac{1}{\sqrt{a+1}}+\frac{3}{\sqrt{b+3}}+\frac{3}{\sqrt{c+3}}=\frac72\] Prove that $abc\leq 3$.\\ I was asked to propose a inequality for the condition of $abc<3$ inequality since <3 looks like a heart shape, then I construct a equality and with the help of wolfram, I gave the birth of this bad-looking inequality, I’m glad to see any method besides calculus.

1985 AIME Problems, 10
Tags: floor function , calculus , integration , function
How many of the first 1000 positive integers can be expressed in the form \[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \] where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

2010 USA Team Selection Test, 1
Tags: number theory , greatest common divisor , algebra , polynomial , modular arithmetic , induction , calculus
Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and \[\gcd(P(0), P(1), P(2), \ldots ) = 1.\] Show there are infinitely many $n$ such that \[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

1962 Miklós Schweitzer, 5
Tags: function , calculus , derivative , real analysis
Let $ f$ be a finite real function of one variable. Let $ \overline{D}f$ and $ \underline{D}f$ be its upper and lower derivatives, respectively, that is, \[ \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}\] , \[ \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}.\] Show that $ \overline{D}f$ and $ \underline{D}f$ are Borel-measurable functions. [A. Csaszar]

2007 Today's Calculation Of Integral, 175
Tags: calculus , integration , calculus computations , logarithm
Evaluate $\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.$

2012 Today's Calculation Of Integral, 783
Tags: calculus , integration , calculus computations
Define a sequence $a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots)$. (1) Find $a_n$. (2) Let ${b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}})$ for $k=1,\ 2,\ 3,\ \cdots$. Prove that $\sum_{k=1}^n b_k<\sqrt{2}-1$ for each $n$. Last Edited

1988 Flanders Math Olympiad, 4
Tags: calculus , derivative , geometry , trigonometry , function , inequalities , quadratic
Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad). Prove $2R\theta$ is between $1$ and $\pi$.

2009 Today's Calculation Of Integral, 475
Tags: calculus , integration , calculus computations
For a positive constant number $ t$, let denote $ D$ the region surrounded by the curve $ y \equal{} e^{x}$, the line $ x \equal{} t$, the $ x$ axis and the $ y$ axis. Let $ V_x,\ V_y$ be the volumes of the solid obtained by rotating $ D$ about the $ x$ axis and the $ y$ axis respectively. Compare the size of $ V_x,\ V_y.$

2011 Today's Calculation Of Integral, 733
Tags: calculus , integration , limit , calculus computations
Find $\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.$

2020 Jozsef Wildt International Math Competition, W50
Tags: calculus , integration , inequalities
Let $f:[0,1]\to\mathbb R$ be a differentiable function, while $f'$ is continuous on $[0,1]$ and $|f'(x)|\le1$, $(\forall)x\in[0,1]$. If $$2\left|\int^1_0f(x)dx\right|\le1$$ Show that: $$(n+2)\left|\int^1_0x^nf(x)dx\right|\le1,~(\forall)x\ge1$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

2007 IberoAmerican Olympiad For University Students, 3
Tags: calculus , integration , inequalities , function , real analysis , integral inequality , logarithm
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds: $\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$, where $T$ is the period of $f(x)$.

2007 QEDMO 5th, 5
Tags: induction , algebra , function , domain , calculus , integration , polynomial
Let $ a$, $ b$, $ c$ be three integers. Prove that there exist six integers $ x$, $ y$, $ z$, $ x^{\prime}$, $ y^{\prime}$, $ z^{\prime}$ such that $ a\equal{}yz^{\prime}\minus{}zy^{\prime};\ \ \ \ \ \ \ \ \ \ b\equal{}zx^{\prime}\minus{}xz^{\prime};\ \ \ \ \ \ \ \ \ \ c\equal{}xy^{\prime}\minus{}yx^{\prime}$.

2012 Turkey Junior National Olympiad, 1
Tags: quadratic , calculus , integration , diophantine equation , number theory
Let $x, y$ be integers and $p$ be a prime for which \[ x^2-3xy+p^2y^2=12p \] Find all triples $(x,y,p)$.

2006 Putnam, A1
Tags: analytic geometry , calculus , integration , trigonometry , geometry , projective geometry
Find the volume of the region of points $(x,y,z)$ such that \[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]

2010 Today's Calculation Of Integral, 639
Tags: calculus , integration , calculus computations
Evaluate $\int_0^1 (x+3)\sqrt{xe^x}\ dx.$

1979 AMC 12/AHSME, 11
Tags: calculus , integration
Find a positive integral solution to the equation \[\frac{1+3+5+\dots+(2n-1)}{2+4+6+\dots+2n}=\frac{115}{116}\] $\textbf{(A) }110\qquad\textbf{(B) }115\qquad\textbf{(C) }116\qquad\textbf{(D) }231\qquad\textbf{(E) }\text{The equation has no positive integral solutions.}$

1983 Putnam, A6
Tags: limit , calculus , integration , real analysis
Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

2014 Online Math Open Problems, 17
Tags: trigonometry , geometry , calculus , analytic geometry , derivative , pythagorean theorem
Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$. [i]Proposed by Evan Chen[/i]

1991 Arnold's Trivium, 21
Tags: calculus , derivative , integration
Find the derivative of the solution of the equation $\ddot{x} = \dot{x}^2 + x^3$ with initial condition $x(0) = 0$, $\dot{x}(0) = A$ with respect to $A$ for $A = 0$.

2022 CMIMC Integration Bee, 13
Tags: calculus , integration
\[\int_{-\infty}^\infty e^{-x^2-4/x^2}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2005 Today's Calculation Of Integral, 34
Tags: calculus , integration , probability , calculus computations
Let $p$ be a constant number such that $0<p<1$. Evaluate \[\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}\]

2014 Contests, 2
Tags: calculus , integration , number theory
For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.