Olympiad problems

2005 Today's Calculation Of Integral, 24

Find the minimum value of $\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx$.

2013 Today's Calculation Of Integral, 871

Tags: integration , limit , calculus , trigonometry , calculus computations , definite integral , logarithm

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

2005 Today's Calculation Of Integral, 80

Tags: calculus , function , algebra , integration , ratio , domain , calculus computations

Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

2010 Today's Calculation Of Integral, 526

Tags: calculus , integration , function , calculus computations

For a function satisfying $ f'(x) > 0$ for $ a\leq x\leq b$, let $ F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt$. For what value of $ x$ is $ F(x)$ is minimized?

2013 Today's Calculation Of Integral, 899

Tags: calculus , calculus computations , integration , limit

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2009 Today's Calculation Of Integral, 467

Tags: geometry , calculus , calculus computations , integration , rotation , trigonometry , geometric transformation

Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$. (1) Find the maximum value of $ y$. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis. (3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.

2011 Today's Calculation Of Integral, 728

Tags: calculus , integration , trigonometry , logarithm , calculus computations

Evaluate \[\int_{\frac {\pi}{12}}^{\frac{\pi}{6}} \frac{\sin x-\cos x-x(\sin x+\cos x)+1}{x^2-x(\sin x+\cos x)+\sin x\cos x}\ dx.\]

2010 Today's Calculation Of Integral, 664

Tags: calculus , calculus computations , integration , trigonometry

For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$. Find $I_1+I_2+I_3+I_4$. [i]1992 University of Fukui entrance exam/Medicine[/i]

2007 Today's Calculation Of Integral, 250

Tags: calculus , integration , limit , trigonometry , calculus computations

For a positive constant number $ p$, find $ \lim_{n\to\infty} \frac {1}{n^{p \plus{} 1}}\sum_{k \equal{} 0}^{n \minus{} 1} \int_{2k\pi}^{(2k \plus{} 1)\pi} x^p\sin ^ 3 x\cos ^ 2x\ dx.$

2007 ISI B.Stat Entrance Exam, 2

Tags: function , derivative , calculus , calculus computations , trigonometry

Use calculus to find the behaviour of the function \[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\] and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.

2012 Today's Calculation Of Integral, 855

Tags: function , calculus , integration , calculus computations , limit

Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

2005 Today's Calculation Of Integral, 42

Tags: logarithm , trigonometry , integration , calculus , limit , calculus computations

Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

2012 Today's Calculation Of Integral, 820

Tags: calculus computations , calculus , integration , analytic geometry , limit , logarithm

Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$

2007 Today's Calculation Of Integral, 207

Tags: logarithm , calculus , integration , calculus computations

Evaluate the following definite integral. \[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]

2014 Contests, 903

Tags: real analysis , calculus , integration , function , geometric series , calculus computations

Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$

2012 Today's Calculation Of Integral, 856

Tags: integration , geometry , calculus , analytic geometry , trigonometry , calculus computations

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 $a>0$.

2010 Today's Calculation Of Integral, 606

Tags: geometry , calculus , integration , calculus computations

Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis. 1956 Tokyo Institute of Technology entrance exam

2013 District Olympiad, 4

Tags: calculus computations , function , calculus , limit

Let$f:\mathbb{R}\to \mathbb{R}$be a monotone function. a) Prove that$f$ have side limits in each point ${{x}_{0}}\in \mathbb{R}$. b) We define the function $g:\mathbb{R}\to \mathbb{R}$, $g\left( x \right)=\underset{t\nearrow x}{\mathop{\lim }}\,f\left( t \right)$( $g\left( x \right)$ with limit at at left in $x$). Prove that if the $g$ function is continuous, than the function $f$ is continuous.

2007 Today's Calculation Of Integral, 212

Tags: integration , trigonometry , calculus , calculus computations

For integers $k\ (0\leq k\leq 5)$, positive numbers $m,\ n$ and real numbers $a,\ b$, let $f(k)=\int_{-\pi}^{\pi}(\sin kx-a\sin mx-b\sin nx)^{2}\ dx$, $p(k)=\frac{5!}{k!(5-k)!}\left(\frac{1}{2}\right)^{5}, \ E=\sum_{k=0}^{5}p(k)f(k)$. Find the values of $m,\ n,\ a,\ b$ for which $E$ is minimized.