Olympiad problems

2007 IMS, 5

Find all real $\alpha,\beta$ such that the following limit exists and is finite: \[\lim_{x,y\rightarrow 0^{+}}\frac{x^{2\alpha}y^{2\beta}}{x^{2\alpha}+y^{3\beta}}\]

2010 Today's Calculation Of Integral, 657

Tags: integration , calculus , trigonometry , limit , function , linear algebra , calculus computations

A sequence $a_n$ is defined by $\int_{a_n}^{a_{n+1}} (1+|\sin x|)dx=(n+1)^2\ (n=1,\ 2,\ \cdots),\ a_1=0$. Find $\lim_{n\to\infty} \frac{a_n}{n^3}$.

2008 Teodor Topan, 3

Tags: geometric transformation , geometry , calculus , limit , logarithm , calculus computations

Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.

2011 Today's Calculation Of Integral, 716

Tags: calculus computations , logarithm , calculus , integration

Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}\]

2007 Today's Calculation Of Integral, 201

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

Evaluate the following definite integral. \[\int_{-1}^{1}\frac{e^{2x}+1-(x+1)(e^{x}+e^{-x})}{x(e^{x}-1)}dx\]

2010 Today's Calculation Of Integral, 530

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

Answer the following questions. (1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$. (2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$. (3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.

2005 Today's Calculation Of Integral, 83

Tags: integration , calculus computations , calculus , trigonometry

Evaluate \[\sum_{n=1}^{\infty} \int_{2n\pi}^{2(n+1)\pi} \frac{x\sin x+\cos x}{x^2}\ dx\ (n=1,2,\cdots)\]

2011 Today's Calculation Of Integral, 695

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

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 $\lim_{n\to\infty} T_n.$

2007 Today's Calculation Of Integral, 248

Tags: trig identities , calculus computations , integration , function , calculus , trigonometry

Evaluate $ \int_{\frac {\pi}{4}}^{\frac {3}{4}\pi } \cos \frac {1}{\sin \left(\frac {1}{\sin x}\right)}\cdot \cos \left(\frac {1}{\sin x}\right)\cdot \frac {\cos x}{\sin ^ 2 x\cdot \sin ^ 2 \left(\frac {1}{\sin x }\right)}\ dx$ Last Edited, Sorry kunny

2005 Today's Calculation Of Integral, 72

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

Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$ Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.

2013 Waseda University Entrance Examination, 4

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

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

2010 Today's Calculation Of Integral, 660

Tags: calculus computations , integration , logarithm , calculus

Let $a,\ b$ be given positive constants. Evaluate \[\int_0^1 \frac{\ln\ (x+a)^{x+a}(x+b)^{x+b}}{(x+a)(x+b)}dx.\] Own

2010 Today's Calculation Of Integral, 543

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

Let $ y$ be the function of $ x$ satisfying the differential equation $ y'' \minus{} y \equal{} 2\sin x$. (1) Let $ y \equal{} e^xu \minus{} \sin x$, find the differential equation with which the function $ u$ with respect to $ x$ satisfies. (2) If $ y(0) \equal{} 3,\ y'(0) \equal{} 0$, then determine $ y$.

2005 Today's Calculation Of Integral, 39

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

Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$. \[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]

2010 Today's Calculation Of Integral, 542

Tags: function , calculus , integration , calculus computations

Find continuous functions $ f(x),\ g(x)$ which takes positive value for any real number $ x$, satisfying $ g(x)\equal{}\int_0^x f(t)\ dt$ and $ \{f(x)\}^2\minus{}\{g(x)\}^2\equal{}1$.

2005 Today's Calculation Of Integral, 87

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

Find the minimum value of $a\ (0<a<1)$ for which the following definite integral is minimized. \[ \int_0^{\pi} |\sin x-ax|\ dx \]

2012 Kyoto University Entry Examination, 1A

Tags: calculus , calculus computations , geometry , symmetry , integration

Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.

2013 Today's Calculation Of Integral, 875

Tags: calculus , trigonometry , integration , ratio , calculus computations

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$

2012 Today's Calculation Of Integral, 838

Tags: calculus , integration , calculus computations , inequalities

Prove that : $\frac{e-1}{e}<\int_0^1 e^{-x^2}dx<\frac{\pi}{4}.$

2009 Today's Calculation Of Integral, 436

Tags: calculus , calculus computations , integration , geometry

Find the minimum area bounded by the graphs of $ y\equal{}x^2$ and $ y\equal{}kx(x^2\minus{}k)\ (k>0)$.

2007 Today's Calculation Of Integral, 197

Tags: function , trigonometry , integration , algebra , calculus , rational function , calculus computations

Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral. \[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]

2012 Today's Calculation Of Integral, 779

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

Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane. When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.

2008 Harvard-MIT Mathematics Tournament, 6

Tags: calculus , limit , calculus computations , ratio

Determine the value of $ \lim_{n\rightarrow\infty}\sum_{k \equal{} 0}^n\binom{n}{k}^{ \minus{} 1}$.

1990 Flanders Math Olympiad, 4

Tags: function , calculus , limit , calculus computations

Let $f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0$ be a strictly decreasing function. (a) Be $a_n$ a sequence of strictly positive reals so that $\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1})$ Prove that $a_n$ is ascending, that $\displaystyle\lim_{k\rightarrow +\infty} f(a_k)$ = 0and that $\displaystyle\lim_{k\rightarrow +\infty} a_k =+\infty$ (b) Prove that there exist such a sequence ($a_n$) in $\mathbb{R}^+_0$ if you know $\displaystyle\lim_{x\rightarrow +\infty} f(x)=0$.

2011 Today's Calculation Of Integral, 711

Tags: calculus , integration , calculus computations , logarithm

Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$