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: 2215

2002 AIME Problems, 10
Tags: trigonometry , calculus
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m\pi}{n-\pi}$ and $\frac{p\pi}{q+\pi},$ where $m,$ $n,$ $p$ and $q$ are positive integers. Find $m+n+p+q.$

2018 ISI Entrance Examination, 4
Tags: calculus
Let $f:(0,\infty)\to\mathbb{R}$ be a continuous function such that for all $x\in(0,\infty)$, $$f(2x)=f(x)$$ Show that the function $g$ defined by the equation $$g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0$$ is a constant function.

2001 Vietnam National Olympiad, 3
Tags: trigonometry , integration , calculus , calculus computations
For real $a, b$ define the sequence $x_{0}, x_{1}, x_{2}, ...$ by $x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}$. If $b = 1$, show that the sequence converges to a finite limit for all $a$. If $b > 2$, show that the sequence diverges for some $a$.

2009 Iran Team Selection Test, 3
Tags: calculus , function , three variable inequality , algebra , inequality
Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that : $ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$

2004 Germany Team Selection Test, 1
Tags: calculus , inequalities , bounded , algebra , sequence
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.

2020 ISI Entrance Examination, 6
Tags: calculus
Prove that the family of curves $$\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1$$ satisfies $$\frac{dy}{dx}(a^2-b^2)=\left(x+y\frac{dy}{dx}\right)\left(x\frac{dy}{dx}-y\right)$$

2021 CMIMC Integration Bee, 7
Tags: calculus , integration
$$\int_0^\infty \frac{1}{(x^2+4)^{5/2}}\,dx$$ [i]Proposed by Connor Gordon[/i]

2012 Hanoi Open Mathematics Competitions, 7
Tags: calculus , integration
[b]Q7.[/b] Find all integers $n$ such that $60+2n-n^2$ is a perfect square.

2005 Today's Calculation Of Integral, 6
Tags: calculus , integration , trigonometry , derivative , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int \sin x\cos ^ 3 x dx$ [2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$ [3] $\int x^2 \sqrt{x^3+1}dx$ [4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$ [5] $\int (1-x^2)e^x dx$

2021 JHMT HS, 10
Tags: algebra , polynomial , calculus , derivative
A polynomial $P(x)$ of some degree $d$ satisfies $P(n) = n^3 + 10n^2 - 12$ and $P'(n) = 3n^2 + 20n - 1$ for $n = -2, -1, 0, 1, 2.$ Also, $P$ has $d$ distinct (not necessarily real) roots $r_1, r_2, \ldots, r_d.$ The value of \[ \sum_{k=1}^{d}\frac{1}{4 - r_k^2} \] can be expressed as a common fraction $\tfrac{p}{q}.$ What is the value of $p + q?$

2002 VJIMC, Problem 1
Tags: linear algebra , function , calculus , derivative , linear independence
Differentiable functions $f_1,\ldots,f_n:\mathbb R\to\mathbb R$ are linearly independent. Prove that there exist at least $n-1$ linearly independent functions among $f_1',\ldots,f_n'$.

1976 USAMO, 2
Tags: analytic geometry , graphing lines , slope , function , calculus , derivative
If $ A$ and $ B$ are fixed points on a given circle and $ XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $ AX$ and $ BY$. You may assume that $ AB$ is not a diameter.

2004 Olympic Revenge, 5
Tags: calculus , integration , induction , number theory
$a_0 = a_1 = 1$ and ${a_{n+1} . a_{n-1}} = a_n . (a_n + 1)$ for all positive integers n. prove that $a_n$ is one integer for all positive integers n.

2018 Moscow Mathematical Olympiad, 1
Tags: calculus , derivative , algebra
The graphs of a square trinomial and its derivative divide the coordinate plane into four parts. How many roots does this square trinomial has?

Today's calculation of integrals, 876
Tags: function , calculus , integration , calculus computations , definite integral
Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

2011 China Team Selection Test, 2
Tags: calculus , function , number theory , combinatorics
Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$. Show that for all positive integers $r$, \[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]

1967 IMO Shortlist, 3
Tags: trigonometry , calculus , taylor series , inequality , trigonometric inequality
Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$

2012 Today's Calculation Of Integral, 843
Tags: calculus , integration , function , inequalities , special factorizations , calculus computations
Let $f(x)$ be a continuous function such that $\int_0^1 f(x)\ dx=1.$ Find $f(x)$ for which $\int_0^1 (x^2+x+1)f(x)^2dx$ is minimized.

2005 Today's Calculation Of Integral, 5
Tags: calculus , integration , trigonometry , calculus computations
Calculate the following indefinite integrals. [1] $\int (4-5\tan x)\cos x dx$ [2] $\int \frac{dx}{\sqrt[3]{(1-3x)^2}}dx$ [3] $\int x^3\sqrt{4-x^2}dx$ [4] $\int e^{-x}\sin \left(x+\frac{\pi}{4}\right)dx$ [5] $\int (3x-4)^2 dx$

2010 Today's Calculation Of Integral, 571
Tags: calculus , integration , trigonometry , function , calculus computations , logarithm
Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.

Oliforum Contest I 2008, 3
Tags: calculus , derivative , symmetry , inequalities
Let $ a,b,c$ be three pairwise distinct real numbers such that $ a\plus{}b\plus{}c\equal{}6\equal{}ab\plus{}bc\plus{}ca\minus{}3$. Prove that $ 0<abc<4$.

2005 VTRMC, Problem 6
Tags: integration , calculus
Compute $\int^1_0\left((e-1)\sqrt{\ln(1+ex-x)}+e^{x^2}\right)dx$.

2005 Today's Calculation Of Integral, 71
Tags: calculus , integration , calculus computations
Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$

2007 Today's Calculation Of Integral, 201
Tags: calculus , integration , limit , function , calculus computations
Evaluate the following definite integral. \[\int_{-1}^{1}\frac{e^{2x}+1-(x+1)(e^{x}+e^{-x})}{x(e^{x}-1)}dx\]

2005 Vietnam Team Selection Test, 1
Tags: calculus , inequalities , three variable inequality
Let be given positive reals $a$, $b$, $c$. Prove that: $\frac{a^{3}}{\left(a+b\right)^{3}}+\frac{b^{3}}{\left(b+c\right)^{3}}+\frac{c^{3}}{\left(c+a\right)^{3}}\geq \frac{3}{8}$.