Olympiad problems

Today's calculation of integrals, 864

Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

2009 Today's Calculation Of Integral, 437

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.$

2005 Today's Calculation Of Integral, 31

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

Evaluate \[\lim_{n\to\infty} \int_0^{\pi} x^2 |\sin nx| dx\]

2010 Today's Calculation Of Integral, 664

Tags: calculus , integration , trigonometry , calculus computations

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]

Today's calculation of integrals, 895

Tags: calculus , integration , analytic geometry , parabola , calculus computations , conic , geometry

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.

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 $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

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 \[ \int_0^{\pi} (\sqrt[2009]{\cos x}\plus{}\sqrt[2009]{\sin x}\plus{}\sqrt[2009]{\tan x})\ dx.\]

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)$.

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 $\int_1^e \left|\ln x-\frac{a}{x}\right|dx\ (0\leq a\leq e)$

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]

2011 Today's Calculation Of Integral, 724

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

Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.

2011 Today's Calculation Of Integral, 693

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\pi} \sqrt[4]{1+|\cos x|}\ dx.$ created by kunny

2009 Today's Calculation Of Integral, 439

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

Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$. Note that you may not directively use double integral here for Japanese high school students who don't study it.

2011 Tokyo Instutute Of Technology Entrance Examination, 1

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

Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$. For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$