Olympiad problems

2007 Moldova National Olympiad, 12.4

If the function $f\colon [1,2]\to R$ is such that $\int_{1}^{2}f(x) dx=\frac{73}{24}$, then show that there exists a $x_{0}\in (1;2)$ such that \[x_{0}^{2}<f(x_{0})<x_{0}^{3}\] [Edit: $f$ is continuous]

2007 Today's Calculation Of Integral, 171

Tags: calculus computations , integration , calculus

Evaluate $\int_{0}^{1}x^{2007}(1-x^{2})^{1003}dx.$

2005 Today's Calculation Of Integral, 40

Tags: calculus computations , integration , calculus

Evaluate \[\int_0^1 x^{2005}e^{-x^2}dx\]

2009 Today's Calculation Of Integral, 419

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

In the $ xy$ plane, the line $ l$ touches to 2 parabolas $ y\equal{}x^2\plus{}ax,\ y\equal{}x^2\minus{}2ax$, where $ a$ is positive constant. (1) Find the equation of $ l$. (2) Find the area $ S$ bounded by the parabolas and the tangent line $ l$.

2010 Today's Calculation Of Integral, 583

Tags: calculus , geometry , integration , calculus computations

Find the values of $ k$ such that the areas of the three parts bounded by the graph of $ y\equal{}\minus{}x^4\plus{}2x^2$ and the line $ y\equal{}k$ are all equal.

2010 Today's Calculation Of Integral, 601

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

Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

2009 Today's Calculation Of Integral, 443

Tags: trigonometry , integration , calculus , calculus computations

Evaluate $ \int_1^{e^2} \frac{(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)\plus{}(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)}{\sqrt{x}}\ dx.$

2007 Today's Calculation Of Integral, 189

Tags: limit , calculus , integration , 3d geometry , calculus computations , geometry

Let $n$ be positive integers. Denote the graph of $y=\sqrt{x}$ by $C,$ and the line passing through two points $(n,\ \sqrt{n})$ and $(n+1,\ \sqrt{n+1})$ by $l.$ Let $V$ be the volume of the solid obtained by revolving the region bounded by $C$ and $l$ around the $x$ axis.Find the positive numbers $a,\ b$ such that $\lim_{n\to\infty}n^{a}V=b.$

2010 Today's Calculation Of Integral, 638

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

Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$ 1978 Chuo university entrance exam/Science and Technology

2010 Today's Calculation Of Integral, 600

Tags: trigonometry , integration , calculus , calculus computations

Evaluate $\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)$. created by kunny

2013 Today's Calculation Of Integral, 863

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

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

2010 Today's Calculation Of Integral, 565

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

Prove that $ f(x)\equal{}\int_0^1 e^{\minus{}|t\minus{}x|}t(1\minus{}t)dt$ has maximal value at $ x\equal{}\frac 12$.

Today's calculation of integrals, 898

Tags: calculus , integration , logarithm , calculus computations

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

2012 Today's Calculation Of Integral, 788

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

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

Today's calculation of integrals, 885

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

Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

2010 Today's Calculation Of Integral, 590

Tags: trigonometry , integration , calculus computations , calculus

Evaluate $ \int_0^{\frac{\pi}{8}} \frac{(\cos \theta \plus{}\sin \theta)^{\frac{3}{2}}\minus{}(\cos \theta \minus{}\sin \theta)^{\frac{3}{2}}}{\sqrt{\cos 2\theta}}\ d\theta$.

2013 Today's Calculation Of Integral, 869

Tags: integration , trigonometry , calculus computations , calculus

Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$ Answer the questions below. (1) Find $I_n.$ (2) Find $\sum_{n=1}^{\infty} I_n.$

2007 Today's Calculation Of Integral, 194

Tags: calculus , integration , calculus computations

Evaluate \[\sum_{n=0}^{2006}\int_{0}^{1}\frac{dx}{2(x+n+1)\sqrt{(x+n)(x+n+1)}}\]

2012 Today's Calculation Of Integral, 851

Tags: calculus computations , limit , function , calculus , integration , geometric series , trigonometry

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

2010 Today's Calculation Of Integral, 623

Tags: calculus , function , integration , calculus computations

Find the continuous function satisfying the following equation. \[\int_0^x f(t)dt+\int_0^x tf(x-t)dt=e^{-x}-1.\] [i]1978 Shibaura Institute of Technology entrance exam[/i]

2009 Today's Calculation Of Integral, 454

Tags: trigonometry , calculus , calculus computations , integration

Let $ a$ be positive constant number. Evaluate $ \int_{ \minus{} a}^a \frac {x^2\cos x \plus{} e^{x}}{e^{x} \plus{} 1}\ dx.$

Today's calculation of integrals, 767

Tags: integration , calculus , calculus computations , trigonometry

For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

2007 Today's Calculation Of Integral, 238

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

Find $ \lim_{a\to\infty} \frac {1}{a^2}\int_0^a \log (1 \plus{} e^x)\ dx.$

2005 Today's Calculation Of Integral, 61

Tags: trigonometry , calculus , calculus computations , integration

Evaluate \[\sum_{k=0}^{2004} \int_{-1}^1 \frac{\sqrt{1-x^2}}{\sqrt{k+1}-x}dx\]

2009 Today's Calculation Of Integral, 517

Tags: geometry , integration , calculus , search , calculus computations

Consider points $ P$ which are inside the square with side length $ a$ such that the distance from $ P$ to the center of the square equals to the least distance from $ P$ to each side of the square.Find the area of the figure formed by the whole points $ P$.