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: 1687

2019 Ramnicean Hope, 2
Tags: calculus , integration , definite integral , quadratic
Calculate $ \int_1^4 \frac{\ln x}{(1+x)(4+x)} dx . $ [i]Ovidiu Țâțan[/i]

2005 All-Russian Olympiad, 2
Tags: calculus , integration , quadratic , algebra
Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty).

2022 CMIMC Integration Bee, 12
Tags: calculus , integration
\[\int_{\pi/4}^{\pi/2} \tan^{-1}\left(\tan^2(x)\right)\sin(2x)\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2007 Moldova National Olympiad, 12.8
Tags: calculus , integration , function , real analysis , inequalities
Find all continuous functions $f\colon [0;1] \to R$ such that \[\int_{0}^{1}f(x)dx = 2\int_{0}^{1}(f(x^{4}))^{2}dx+\frac{2}{7}\]

Today's calculation of integrals, 870
Tags: calculus , integration , ellipse , hyperbola , geometry , inequalities , conic
Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]

1990 National High School Mathematics League, 10
Tags: calculus , integration
Define $f(n):$ the number of integral points of line segment $OA_n$ ($O$ and $A_n$ not included), where $A_n(n,n+3)$. Then, $f(1)+f(2)+\cdots+f(1990)=$________.

2007 Today's Calculation Of Integral, 194
Tags: calculus , integration , calculus computations
Evaluate \[\sum_{n=0}^{2006}\int_{0}^{1}\frac{dx}{2(x+n+1)\sqrt{(x+n)(x+n+1)}}\]

Today's calculation of integrals, 876
Tags: calculus , integration , calculus computations , definite integral , function
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 $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

2010 IMC, 2
Tags: integration , real analysis , infinite series , logarithm
Compute the sum of the series $\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$

2007 Today's Calculation Of Integral, 240
Tags: calculus , integration , geometry , derivative , analytic geometry , calculus computations
2 curves $ y \equal{} x^3 \minus{} x$ and $ y \equal{} x^2 \minus{} a$ pass through the point $ P$ and have a common tangent line at $ P$. Find the area of the region bounded by these curves.

2009 Today's Calculation Of Integral, 499
Tags: calculus , integration , trigonometry , symmetry , function , calculus computations
Evaluate \[ \int_0^{\pi} (\sqrt[2009]{\cos x}\plus{}\sqrt[2009]{\sin x}\plus{}\sqrt[2009]{\tan x})\ dx.\]

2000 CentroAmerican, 1
Tags: calculus , integration
Find all three-digit numbers $ abc$ (with $ a \neq 0$) such that $ a^{2}+b^{2}+c^{2}$ is a divisor of 26.

2007 Today's Calculation Of Integral, 249
Tags: calculus , integration , calculus computations , logarithm
Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.

2002 District Olympiad, 3
Tags: calculus , integration , real analysis , logarithm
[b]a)[/b] Calculate $ \lim_{n\to\infty} \int_0^{\alpha } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx , $ for all $ \alpha\in (0,1) . $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \int_0^{1 } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx . $

2010 Today's Calculation Of Integral, 549
Tags: calculus , integration , function , algebra , polynomial , limit , calculus computations
Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.

2005 Today's Calculation Of Integral, 68
Tags: calculus , integration , derivative , calculus computations , logarithm
Find the minimum value of $\int_1^e \left|\ln x-\frac{a}{x}\right|dx\ (0\leq a\leq e)$

2017 BMT Spring, 3
Tags: algebra , calculus , integration
Compute $\int^9_{-9}17x^3 \cos (x^2) dx.$

2010 Morocco TST, 2
Tags: integration , floor function , calculus , algebra , series summation , approximation
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$

2009 Today's Calculation Of Integral, 423
Tags: calculus , integration , analytic geometry , graphing lines , slope , function , absolute value
Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ . Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$.

2009 Harvard-MIT Mathematics Tournament, 8
Tags: calculus , integration , logarithm
Compute \[\int_1^{\sqrt{3}} x^{2x^2+1}+\ln\left(x^{2x^{2x^2+1}}\right)dx.\]

2010 ELMO Shortlist, 4
Tags: integration , combinatorics
The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$. [i]Brian Hamrick.[/i]

2023 CMIMC Integration Bee, 10
Tags: calculus , integration
\[\int_{\frac 1{\sqrt 3}}^{\sqrt 3} \frac{\arctan(x)\log^2(x)}{x}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2009 Today's Calculation Of Integral, 509
Tags: calculus , integration , trigonometry , algebra , partial fractions , calculus computations , logarithm
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.

2010 Today's Calculation Of Integral, 666
Tags: calculus , integration , function , trigonometry , calculus computations , logarithm
Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying: (i) $f\left(\frac{\pi}{6}\right)=0$ (ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$. Find $f(x)$. [i]1987 Sapporo Medical University entrance exam[/i]

1980 IMO, 8
Tags: calculus , integration , number theory
Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).