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

2007 Today's Calculation Of Integral, 230
Tags: calculus , integration , function , calculus computations , logarithm
Prove that $ \frac {( \minus{} 1)^n}{n!}\int_1^2 (\ln x)^n\ dx \equal{} 2\sum_{k \equal{} 1}^n \frac {( \minus{} \ln 2)^k}{k!} \plus{} 1$.

2005 Today's Calculation Of Integral, 62
Tags: calculus , integration , limit , calculus computations , logarithm
For $a>1$, let $f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)$ and let $b_n$ be the minimum value of $f(a)$ at $a>1$. Evaluate \[\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)\]

2013 Today's Calculation Of Integral, 893
Tags: calculus , integration , trigonometry , function , derivative , calculus computations , logarithm
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

2010 Today's Calculation Of Integral, 582
Tags: calculus , integration , trigonometry , inequalities , calculus computations
Prove the following inequality. \[ \frac{\pi}{4}\sqrt{\frac{3}{2}\plus{}\sqrt{2}}<\int_0^{\frac{\pi}{2}} \sqrt{1\minus{}\frac 12\sin ^ 2 x}\ dx<\frac{\sqrt{3}}{4}\pi\]

2010 Today's Calculation Of Integral, 536
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx$.

2007 Today's Calculation Of Integral, 232
Tags: calculus , integration , trigonometry , inequalities , calculus computations
For $ f(x)\equal{}1\minus{}\sin x$, let $ g(x)\equal{}\int_0^x (x\minus{}t)f(t)\ dt.$ Show that $ g(x\plus{}y)\plus{}g(x\minus{}y)\geq 2g(x)$ for any real numbers $ x,\ y.$

2010 Today's Calculation Of Integral, 597
Tags: calculus , integration , calculus computations
In space given a board shaped the equilateral triangle $PQR$ with vertices $P\left(1,\ \frac 12,\ 0\right),\ Q\left(1,-\frac 12,\ 0\right),\ R\left(\frac 14,\ 0,\ \frac{\sqrt{3}}{4}\right)$. When $S$ is revolved about the $z$-axis, find the volume of the solid generated by the whole points through which $S$ passes. 1984 Tokyo University entrance exam/Science

2007 Today's Calculation Of Integral, 211
Tags: calculus , integration , parabola , hyperbola , geometry , calculus computations , conic
When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves, prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.

2008 Harvard-MIT Mathematics Tournament, 9
Tags: limit , integration , hmmt , calculus , real analysis , calculus computations , logarithm
([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.

2005 Today's Calculation Of Integral, 55
Tags: calculus , integration , limit , function , calculus computations
Evaluate \[\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( n=1,2,\cdots)\]

2010 Today's Calculation Of Integral, 624
Tags: calculus , integration , function , trigonometry , real analysis , calculus computations
Find the continuous function $f(x)$ such that the following equation holds for any real number $x$. \[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\] [i]1977 Keio University entrance exam/Medicine[/i]

2011 Today's Calculation Of Integral, 705
Tags: calculus , integration , trigonometry , calculus computations
The parametric equations of a curve are given by $x = 2(1+\cos t)\cos t,\ y =2(1+\cos t)\sin t\ (0\leq t\leq 2\pi)$. (1) Find the maximum and minimum values of $x$. (2) Find the volume of the solid enclosed by the figure of revolution about the $x$-axis.

2011 Today's Calculation Of Integral, 741
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\int_0^1 \frac{(x-1)^2(\cos x+1)-(2x-1)\sin x}{(x-1+\sqrt{\sin x})^2}\ dx\]

2009 Today's Calculation Of Integral, 486
Tags: calculus , integration , trigonometry , analytic geometry , calculus computations , geometry
Let $ H$ be the piont of midpoint of the cord $ PQ$ that is on the circle centered the origin $ O$ with radius $ 1.$ Suppose the length of the cord $ PQ$ is $ 2\sin \frac {t}{2}$ for the angle $ t\ (0\leq t\leq \pi)$ that is formed by half-ray $ OH$ and the positive direction of the $ x$ axis. Answer the following questions. (1) Express the coordiante of $ H$ in terms of $ t$. (2) When $ t$ moves in the range of $ 0\leq t\leq \pi$, find the minimum value of $ x$ coordinate of $ H$. (3) When $ t$ moves in the range of $ 0\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the region bounded by the curve drawn by the point $ H$ and the $ x$ axis and the $ y$ axis.

2012 Today's Calculation Of Integral, 813
Tags: calculus , integration , geometry , inequalities , quadratic , calculus computations
Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$. How many solutions (including University Mathematics )are there for the problem? Any advice would be appreciated.

2013 Today's Calculation Of Integral, 874
Tags: calculus , integration , parabola , geometry , calculus computations , conic
Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2014 Contests, 902
Tags: calculus , integration , calculus computations
For $a\geq 0$, find the minimum value of $S(a)=\int_0^1 |x^2+2ax+a^2-1|\ dx.$

2011 Today's Calculation Of Integral, 683
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $\int_0^{\frac 12} (x+1)\sqrt{1-2x^2}\ dx$. [i]2011 Kyoto University entrance exam/Science, Problem 1B[/i]

2012 Today's Calculation Of Integral, 851
Tags: calculus , integration , function , trigonometry , limit , geometric series , calculus computations
Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

2008 Harvard-MIT Mathematics Tournament, 7
Tags: geometry , 3d geometry , limit , function , calculus , derivative , calculus computations
([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.

2012 Today's Calculation Of Integral, 856
Tags: calculus , integration , analytic geometry , trigonometry , calculus computations , geometry
On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

Today's calculation of integrals, 889
Tags: calculus , integration , geometry , calculus computations , logarithm
Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

2009 Today's Calculation Of Integral, 403
Tags: calculus , integration , calculus computations
Evaluate $ \int_0^1 \frac{2e^{2x}\plus{}xe^x\plus{}3e^x\plus{}1}{(e^x\plus{}1)^2(e^x\plus{}x\plus{}1)^2}\ dx$.

2011 Today's Calculation Of Integral, 709
Tags: calculus , integration , trigonometry , function , calculus computations
Evaluate $ \int_0^1 \frac{x}{1\plus{}x}\sqrt{1\minus{}x^2}\ dx$.

2005 Today's Calculation Of Integral, 76
Tags: calculus , integration , function , limit , inequalities , calculus computations
The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows. \[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\] Evaluate \[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]