Olympiad problems

1992 Vietnam National Olympiad, 3

Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.

2007 Today's Calculation Of Integral, 185

Tags: calculus , integration , calculus computations , trigonometry

Evaluate the following integrals. (1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$ (2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$

2012 Today's Calculation Of Integral, 817

Tags: function , logarithm , calculus , geometry , calculus computations , integration

Define two functions $f(t)=\frac 12\left(t+\frac{1}{t}\right),\ g(t)=t^2-2\ln t$. When real number $t$ moves in the range of $t>0$, denote by $C$ the curve by which the point $(f(t),\ g(t))$ draws on the $xy$-plane. Let $a>1$, find the area of the part bounded by the line $x=\frac 12\left(a+\frac{1}{a}\right)$ and the curve $C$.

2013 Today's Calculation Of Integral, 872

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

Let $n$ be a positive integer. (1) For a positive integer $k$ such that $1\leq k\leq n$, Show that : \[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\] (2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$ If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$ (3) Find $\lim_{n\to\infty} S_n.$

2011 Today's Calculation Of Integral, 708

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

Find $ \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots)$.

2007 Moldova National Olympiad, 11.2

Tags: limit , logarithm , calculus , calculus computations

Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$. Find $\lim_{n\rightarrow\infty}a_{n}$.

2010 Today's Calculation Of Integral, 630

Tags: integration , logarithm , calculus , calculus computations

Evaluate $\int_0^{\infty} \frac{\ln (1+e^{4x})}{e^x}dx.$

2012 Today's Calculation Of Integral, 832

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

Find the limit \[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]

2012 Today's Calculation Of Integral, 836

Tags: integration , calculus , trigonometry , calculus computations

Evaluate $\int_0^{\pi} e^{\sin x}\cos ^ 2(\sin x )\cos x\ dx$.

2005 Today's Calculation Of Integral, 86

Tags: trigonometry , integration , calculus , calculus computations

Prove \[\left[\int_\pi^\infty \frac{\cos x}{x}\ dx\right]^2< \frac{1}{{\pi}^2}\]

2005 Today's Calculation Of Integral, 55

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

Evaluate \[\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( n=1,2,\cdots)\]

2012 Today's Calculation Of Integral, 834

Tags: integration , calculus computations , calculus , geometry

Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$.

2005 Today's Calculation Of Integral, 26

Tags: calculus , logarithm , integration , calculus computations

Evaluate \[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]

2013 Today's Calculation Of Integral, 890

Tags: algorithm , calculus , integration , function , calculus computations

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

2009 Today's Calculation Of Integral, 510

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

(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$. (2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.

2005 Today's Calculation Of Integral, 21

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

[1] Tokyo Univ. of Science: $\int \frac{\ln x}{(x+1)^2}dx$ [2] Saitama Univ.: $\int \frac{5}{3\sin x+4\cos x}dx$ [3] Yokohama City Univ.: $\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx$ [4] Daido Institute of Technology: $\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx$ [5] Gunma Univ.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$

2009 Today's Calculation Of Integral, 406

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

Find $ \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos (2n\plus{}1)x|\ dx$.

2007 ISI B.Stat Entrance Exam, 7

Tags: trigonometry , prism , 3d geometry , calculus computations , calculus , geometry

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.

2011 Today's Calculation Of Integral, 764

Tags: calculus , integration , trigonometry , function , 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 $\int_0^{\pi} \{f(x)\}^3dx.$

2010 Today's Calculation Of Integral, 540

Tags: calculus , calculus computations , logarithm , integration

Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.

2005 Today's Calculation Of Integral, 35

Tags: calculus , derivative , function , integration , calculus computations

Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.

2005 Today's Calculation Of Integral, 7

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

Calculate the following indefinite integrals. [1] $\int \sqrt{x}(\sqrt{x}+1)^2 dx$ [2] $\int (e^x+2e^{x+1}-3e^{x+2})dx$ [3] $\int (\sin ^2 x+\cos x)\sin x dx$ [4] $\int x\sqrt{2-x} dx$ [5] $\int x\ln x dx$

2005 Today's Calculation Of Integral, 78

Tags: calculus , integration , calculus computations , trigonometry

Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate \[\int_0^1 \sin \alpha x\sin \beta x\ dx\]

2007 Today's Calculation Of Integral, 242

Tags: geometry , calculus computations , integration , function , calculus

A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$. Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

2009 Today's Calculation Of Integral, 459

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

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)$.