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

1990 AMC 12/AHSME, 7
Tags: calculus , integration , geometry , perimeter
A triangle with integral sides has perimeter $8$. The area of the triangle is $\textbf{(A) }2\sqrt{2}\qquad \textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad \textbf{(C) }2\sqrt{3}\qquad \textbf{(D) }4\qquad \textbf{(E) }4\sqrt{2}$

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4
Tags: parameterization , calculus , integration , logarithm , real analysis
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.

1987 AIME Problems, 11
Tags: calculus , integration , quadratic , inequalities , algorithm , algebra
Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

2005 Today's Calculation Of Integral, 26
Tags: calculus , integration , calculus computations , logarithm
Evaluate \[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]

2004 France Team Selection Test, 1
Tags: inequalities , calculus , derivative , function
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

2012 Centers of Excellency of Suceava, 4
Tags: function , calculus , derivative , real analysis , lagrange , convexity
Let be two real numbers $ a<b $ and a differentiable function $ f:[a,b]\longrightarrow\mathbb{R} $ that has a bounded derivative. Show that if $ \frac{f(b)-f(a)}{b-a} $ is equal to the global supremum or infimum of $ f', $ then $ f $ is polynomial with degree $ 1. $ [i]Cătălin Țigăeru[/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$.

2010 Contests, 2
Tags: circumcircle , geometric transformation , reflection , parallelogram , calculus , trigonometry , geometry
Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram. Show that $\angle BPC > \angle BAC$.

2005 Today's Calculation Of Integral, 22
Tags: calculus , integration , induction , trigonometry , calculus computations
Evaluate \[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]

1986 National High School Mathematics League, 3
Tags: calculus , integration
In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it integral point. Please color all intengral points in white, red and black, satisfying: (1) Points in every color appear on infinitely many lines that are parallel to $x$-axis. (2) For any white point $A$, red point $B$, black point $C$, we can find another red point $D$, such that $ABCD$ is a parallelogram.

2024 CMIMC Integration Bee, 9
Tags: calculus , integration
\[\int_0^1 \frac{1-x}{x^{5/2}+x^{3/2}+x^{1/2}}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2010 Contests, 522
Tags: calculus , integration , limit , function , geometry , derivative , logarithm
Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

2013 Today's Calculation Of Integral, 899
Tags: calculus , integration , limit , calculus computations
Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2004 Vietnam Team Selection Test, 2
Tags: function , calculus , integration , search , algebra
Find all real values of $\alpha$, for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation \[ f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y \] for all $x, y \in \mathbb{R}$.

2011 Today's Calculation Of Integral, 676
Tags: calculus , integration , trigonometry , calculus computations
Let $f(x)=\cos ^ 4 x+3\sin ^ 4 x$. Evaluate $\int_0^{\frac{\pi}{2}} |f'(x)|dx$. [i]2011 Tokyo University of Science entrance exam/Management[/i]

2011 Today's Calculation Of Integral, 746
Tags: calculus , integration , floor function , inequalities , calculus computations
Prove the following inequality. \[n^ne^{-n+1}\leq n!\leq \frac 14(n+1)^{n+1}e^{-n+1}.\]

2007 Today's Calculation Of Integral, 223
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.

2001 USA Team Selection Test, 2
Tags: integration , function , calculus , algebra , binomial theorem , combinatorics
Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

2007 Today's Calculation Of Integral, 185
Tags: calculus , integration , trigonometry , calculus computations
Evaluate the following integrals. (1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$ (2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$

2011 Today's Calculation Of Integral, 716
Tags: calculus , integration , calculus computations , logarithm
Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}\]

2013 Today's Calculation Of Integral, 867
Tags: calculus , integration , quadratic , function , derivative , algebra , linear equation
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2021 Alibaba Global Math Competition, 5
Tags: calculus , complex analysis , ode , differential equation
For the complex-valued function $f(x)$ which is continuous and absolutely integrable on $\mathbb{R}$, define the function $(Sf)(x)$ on $\mathbb{R}$: $(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du$. (a) Find the expression for $S(\frac{1}{1+x^2})$ and $S(\frac{1}{(1+x^2)^2})$. (b) For any integer $k$, let $f_k(x)=(1+x^2)^{-1-k}$. Assume $k\geq 1$, find constant $c_1$, $c_2$ such that the function $y=(Sf_k)(x)$ satisfies the ODE with second order: $xy''+c_1y'+c_2xy=0$.

1965 AMC 12/AHSME, 34
Tags: quadratic , calculus , function
For $ x \ge 0$ the smallest value of $ \frac {4x^2 \plus{} 8x \plus{} 13}{6(1 \plus{} x)}$ is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac {25}{12} \qquad \textbf{(D)}\ \frac {13}{6} \qquad \textbf{(E)}\ \frac {34}{5}$

PEN Q Problems, 2
Tags: algebra , calculus , integration , pigeonhole principle , polynomial
Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.

1982 IMO Shortlist, 15
Tags: inequality , calculus , geometric progression
Show that \[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\] holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.