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

2009 Today's Calculation Of Integral, 494
Tags: calculus , integration , hyperbola , analytic geometry , calculus computations , conic
Suppose the curve $ C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0)$ has a common tangent line at the point $ P$ with the hyperbola $ xy \equal{} 1$ in the first quadrant. (1) Find the value of $ a$ and the coordinate of the point $ P$. (2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment $ OP$ and the curve $ C$ about the line $ OP$. [color=green][Edited.][/color]

1979 IMO Shortlist, 20
Tags: algebra , maximization , optimization , lagrange , calculus
Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$

2020 SEEMOUS, Problem 2
Tags: calculus
Let $k>1$ be a real number. Calculate: (a) $L=\lim_{n\to \infty} \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x.$ (b) $\lim_{n\to \infty} n\left\lbrack L- \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x\right\rbrack.$

2021 The Chinese Mathematics Competition, Problem 6
Tags: calculus
Let $x_1=2021$, $x_n^2-2(x_n+1)x_{n+1}+2021=0$ ($n\geq1$). Prove that the sequence ${x_n}$ converges. Find the limit $\lim_{n \to \infty} x_n$.

2006 VTRMC, Problem 4
Tags: calculus , differential equation
We want to find functions $p(t)$, $q(t)$, $f(t)$ such that (a) $p$ and $q$ are continuous functions on the open interval $(0,\pi)$. (b) $f$ is an infinitely differentiable nonzero function on the whole real line $(-\infty,\infty)$ such that $f(0)=f'(0)=f''(0)$. (c) $y=\sin t$ and $y=f(t)$ are solutions of the differential equation $y''+p(t)y'+q(t)y=0$ on $(0,\pi)$. Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such $f,p,q$.

2007 District Olympiad, 3
Tags: calculus , integration , function , limit , real analysis , inequalities
Find all continuous functions $f : \mathbb R \to \mathbb R$ such that: (a) $\lim_{x \to \infty}f(x)$ exists; (b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.

2006 Flanders Math Olympiad, 4
Tags: function , calculus , algebra , functional equation , system of equations
Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that \[ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. \]

1983 Putnam, B3
Tags: differential equation , calculus
Assume that the differential equation $$y'''+p(x)y''+q(x)y'+r(x)y=0$$has solutions $y_1(x)$, $y_2(x)$, $y_3(x)$ on the real line such that $$y_1(x)^2+y_2(x)^2+y_3(x)^2=1$$for all real $x$. Let $$f(x)=y_1'(x)^2+y_2'(x)^2+y_3'(x)^2.$$Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation $$y'+Ap(x)y=Br(x).$$

2013 Today's Calculation Of Integral, 866
Tags: calculus , integration , geometry , trigonometry , calculus computations
Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

2008 Estonia Team Selection Test, 3
Tags: function , algebra , calculus , inequalities
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2010 Today's Calculation Of Integral, 559
Tags: calculus , integration , geometry , geometric transformation , rotation , trigonometry , calculus computations
In $ xyz$ space, consider two points $ P(1,\ 0,\ 1),\ Q(\minus{}1,\ 1,\ 0).$ Let $ S$ be the surface generated by rotation the line segment $ PQ$ about $ x$ axis. Answer the following questions. (1) Find the volume of the solid bounded by the surface $ S$ and two planes $ x\equal{}1$ and $ x\equal{}\minus{}1$. (2) Find the cross-section of the solid in (1) by the plane $ y\equal{}0$ to sketch the figure on the palne $ y\equal{}0$. (3) Evaluate the definite integral $ \int_0^1 \sqrt{t^2\plus{}1}\ dt$ by substitution $ t\equal{}\frac{e^s\minus{}e^{\minus{}s}}{2}$. Then use this to find the area of (2).

2011 VTRMC, Problem 1
Tags: calculus , integration
Evaluate $\int^4_1\frac{x-2}{(x^2+4)\sqrt x}dx$

MIPT Undergraduate Contest 2019, 1.5 & 2.5
Tags: inequalities , trigonometry , real analysis , calculus
Prove the inequality $$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$ for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$

PEN M Problems, 13
Tags: function , induction , pigeonhole principle , calculus , integration , recursive sequences
The sequence $\{x_{n}\}$ is defined by \[x_{0}\in [0, 1], \; x_{n+1}=1-\vert 1-2 x_{n}\vert.\] Prove that the sequence is periodic if and only if $x_{0}$ is irrational.

2008 All-Russian Olympiad, 5
Tags: calculus , integration , inequalities , number theory
The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 \minus{} ax \plus{} b \equal{} 0$ and $ x^2 \minus{} bx \plus{} a \equal{} 0$ has two integral roots?

2019 Peru Cono Sur TST, P6
Tags: polynomial , algebra , calculus
Two polynomials of the same degree $A(x)=a_nx^n+ \cdots + a_1x+a_0$ and $B(x)=b_nx^n+\cdots+b_1x+b_0$ are called [i]friends[/i] is the coefficients $b_0,b_1, \ldots, b_n$ are a permutation of the coefficients $a_0,a_1, \ldots, a_n$. $P(x)$ and $Q(x)$ be two friendly polynomials with integer coefficients. If $P(16)=3^{2020}$, the smallest possible value of $|Q(3^{2020})|$.

2019 District Olympiad, 4
Tags: improper integral , integral , limit , calculus
Let $a$ be a real number, $a>1.$ Find the real numbers $b \ge 1$ such that $$\lim_{x \to \infty} \int\limits_0^x (1+t^a)^{-b} \mathrm{d}t=1.$$

1997 South africa National Olympiad, 2
Tags: calculus , number theory
Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\dfrac{7}{2}$ times the original one.

1994 China Team Selection Test, 2
Tags: algebra , polynomial , calculus , integration , gauss , number theory , prime number
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

1972 Poland - Second Round, 6
Tags: algebra , calculus
Prove that there exists a function $ f $ defined and differentiable in the set of all real numbers, satisfying the conditions $|f'(x) - f'(y)| \leq 4|x-y|$.

2024 Mexican University Math Olympiad, 6
Tags: polynomial , calculus , derivative , bounded , algebra
Let \( p \) be a monic polynomial with all distinct real roots. Show that there exists \( K \) such that \[ (p(x)^2)'' \leq K(p'(x))^2. \]

2017 Mathematical Talent Reward Programme, MCQ: P 10
Tags: limit , calculus , function , differentiable function
Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $\lim \limits_{x\to \infty}f'(x)=1$, then [list=1] [*] $f$ is increasing [*] $f$ is unbounded [*] $f'$ is bounded [*] All of these [/list]

1984 IMO Longlists, 20
Tags: inequality , three variable inequality , calculus , maximization
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

2011 Federal Competition For Advanced Students, Part 1, 1
Tags: quadratic , modular arithmetic , calculus , integration , prime factorization , number theory
Determine all integer triplets $(x,y,z)$ such that \[x^4+x^2=7^zy^2\mbox{.}\]

2018 Bangladesh Mathematical Olympiad, 7
Tags: calculus , interesting problem , trigonometry , function , geometry
[b]Evaluate[/b] $\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$