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: 713

2010 ISI B.Math Entrance Exam, 8
Tags: function , limit , calculus , calculus computations
Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.

2010 Today's Calculation Of Integral, 632
Tags: calculus , integration , limit , trigonometry , floor function , ceiling function , calculus computations
Find $\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).$ [i]2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam[/i]

2007 Today's Calculation Of Integral, 246
Tags: calculus , integration , algebra , polynomial , function , geometry , calculus computations
An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$

2011 Today's Calculation Of Integral, 764
Tags: calculus , integration , function , trigonometry , inequalities , calculus computations
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2005 Today's Calculation Of Integral, 51
Tags: calculus , integration , function , calculus computations
A function $f(x)$ satisfies $f(x)=f\left(\frac{c}{x}\right)$ for some real number $c(>1)$ and all positive number $x$. If $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$, evaluate $\int_1^c \frac{f(x)}{x} dx$

2014 Contests, 901
Tags: calculus , integration , algebra , polynomial , calculus computations
Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$

2009 Today's Calculation Of Integral, 520
Tags: calculus , integration , calculus computations
Let $ a,\ b,\ c$ be postive constants. Evaluate $ \int_0^1 \frac{2a\plus{}3bx\plus{}4cx^2}{2\sqrt{a\plus{}bx\plus{}cx^2}}\ dx$.

2009 Today's Calculation Of Integral, 471
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_1^e \frac{1\minus{}x(e^x\minus{}1)}{x(1\plus{}xe^x\ln x)}\ dx$.

1994 Vietnam National Olympiad, 3
Tags: trigonometry , calculus , calculus computations
Define the sequence $\{x_{n}\}$ by $x_{0}=a\in (0,1)$ and $x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...)$. Show that the sequence converges and find its limit.

2011 Today's Calculation Of Integral, 695
Tags: calculus , integration , limit , calculus computations , logarithm
For a positive integer $n$, let \[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\] Answer the following questions: (1) Show the following inequality. \[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\] (2) Express $T_n-2S_n$ in terms of $n$. (3) Find the limit $\lim_{n\to\infty} T_n.$

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, 735
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate the following definite integrals: (a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$ (b) $\int_0^{\frac 13} xe^{3x}\ dx$ (c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$ (d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$

2013 Today's Calculation Of Integral, 881
Tags: calculus , integration , trigonometry , calculus computations , definite integral
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

2009 Today's Calculation Of Integral, 459
Tags: calculus , integration , limit , calculus computations , logarithm
Find $ \lim_{x\to\infty} \int_{e^{\minus{}x}}^1 \left(\ln \frac{1}{t}\right)^ n\ dt\ (x\geq 0,\ n\equal{}1,\ 2,\ \cdots)$.

2012 Today's Calculation Of Integral, 853
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

2009 Today's Calculation Of Integral, 448
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.$

2010 Today's Calculation Of Integral, 619
Tags: calculus , integration , function , trigonometry , calculus computations , logarithm
Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized. Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$ [i]2010 Saitama University entrance exam/Mathematics[/i] Last Edited

2005 Today's Calculation Of Integral, 32
Tags: calculus , integration , calculus computations
Evaluate \[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]

2010 Today's Calculation Of Integral, 652
Tags: calculus , integration , calculus computations
Let $a,\ b,\ c$ be positive real numbers such that $b^2>ac.$ Evaluate \[\int_0^{\infty} \frac{dx}{ax^4+2bx^2+c}.\] [i]1981 Tokyo University, Master Course[/i]

2008 Harvard-MIT Mathematics Tournament, 4
Tags: quadratic , limit , calculus , calculus computations , logarithm
([b]4[/b]) Let $ a$, $ b$ be constants such that $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$. Determine the pair $ (a,b)$.

2005 ISI B.Math Entrance Exam, 6
Tags: algebra , polynomial , integration , calculus , geometry , calculus computations
Let $a_0=0<a_1<a_2<...<a_n$ be real numbers . Supppose $p(t)$ is a real valued polynomial of degree $n$ such that $\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1$ Show that , for $0\le j\le n-1$ , the polynomial $p(t)$ has exactly one root in the interval $ (a_j,a_{j+1})$

2011 Today's Calculation Of Integral, 755
Tags: calculus , integration , trigonometry , geometry , geometric transformation , rotation , calculus computations
Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane. Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.

2007 Today's Calculation Of Integral, 207
Tags: calculus , integration , calculus computations , logarithm
Evaluate the following definite integral. \[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]

2007 Today's Calculation Of Integral, 204
Tags: calculus , integration , trigonometry , special factorizations , calculus computations
Evaluate \[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]

Today's calculation of integrals, 882
Tags: calculus , integration , limit , calculus computations , riemann sum , logarithm
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.