Olympiad problems

2009 Today's Calculation Of Integral, 505

In the $ xyz$ space with the origin $ O$, given a cuboid $ K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2$ and the plane $ \alpha : z \equal{} 2$. Draw the perpendicular $ PH$ from $ P$ to the plane. Find the volume of the solid formed by all points of $ P$ which are included in $ K$ such that $ \overline{OP}\leq \overline{PH}$.

2012 Today's Calculation Of Integral, 779

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

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.

2011 Today's Calculation Of Integral, 721

Tags: calculus , integration , function , calculus computations

For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.

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)$.

2011 Today's Calculation Of Integral, 717

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

Let $a_n$ be the area of the part enclosed by the curve $y=x^n\ (n\geq 1)$, the line $x=\frac 12$ and the $x$ axis. Prove that : \[0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}\]

2005 Today's Calculation Of Integral, 44

Tags: calculus , integration , trigonometry , induction , abstract algebra , calculus computations

Evaluate \[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]

2014 Contests, 903

Tags: calculus , integration , function , real analysis , 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 $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$

2005 Today's Calculation Of Integral, 33

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

Evaluate \[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]

2010 Today's Calculation Of Integral, 658

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

Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$ (1) Find the length $L$ of $C$. (2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$. Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School. [i]1997 Kyoto University entrance exam/Science[/i]

Today's calculation of integrals, 886

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

Find the functions $f(x),\ g(x)$ such that $f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$ $g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$

2005 Today's Calculation Of Integral, 56

Tags: limit , integration , calculus , calculus computations

Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{[\sqrt{2n^2-k^2}\ ]}{n^2}\] $[x]$ is the greatest integer $\leq x$.

2005 Today's Calculation Of Integral, 31

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

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

2005 Today's Calculation Of Integral, 28

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\frac{\pi}{4}} \frac{x\cos 5x}{\cos x}dx\]

2009 Today's Calculation Of Integral, 432

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

Define the function $ f(t)\equal{}\int_0^1 (|e^x\minus{}t|\plus{}|e^{2x}\minus{}t|)dx$. Find the minimum value of $ f(t)$ for $ 1\leq t\leq e$.

2012 Today's Calculation Of Integral, 800

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

For a positive constant $a$, find the minimum value of $f(x)=\int_0^{\frac{\pi}{2}} |\sin t-ax\cos t|dt.$

2009 Today's Calculation Of Integral, 398

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

In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality: $ 0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1$ $ x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0$

2007 Today's Calculation Of Integral, 217

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

Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.

2010 Today's Calculation Of Integral, 599

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\frac{\pi}{6}} \frac{e^x(\sin x+\cos x+\cos 3x)}{\cos^ 2 {2x}}\ dx$. created by kunny

2012 Today's Calculation Of Integral, 859

Tags: calculus , integration , geometry , derivative , calculus computations

In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

2010 Today's Calculation Of Integral, 604

Tags: calculus , integration , limit , calculus computations

Let $r$ be a positive integer. Determine the value of $a$ for which the limit value $\lim_{n\to\infty} \frac{\sum_{k=1}^n k^r}{n^a} $ has a non zero finite value, then find the limit value. 1956 Tokyo Institute of Technology entrance exam

2005 Today's Calculation Of Integral, 75

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

A function $f(\theta)$ satisfies the following conditions $(a),(b)$. $(a)\ f(\theta)\geq 0$ $(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1$ Prove the following inequality. \[\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)\]

2007 Today's Calculation Of Integral, 228

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

Let $ x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots)$. (1) Show that $ x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}$. (2) Find the value of $ nx_nx_{n \minus{} 1}$. (3) Show that a sequence $ \{x_n\}$ is monotone decreasing. (4) Find $ \lim_{n\to\infty} nx_n^2$.

2010 Today's Calculation Of Integral, 597

Tags: calculus , integration , calculus computations

In space given a board shaped the equilateral triangle $PQR$ with vertices $P\left(1,\ \frac 12,\ 0\right),\ Q\left(1,-\frac 12,\ 0\right),\ R\left(\frac 14,\ 0,\ \frac{\sqrt{3}}{4}\right)$. When $S$ is revolved about the $z$-axis, find the volume of the solid generated by the whole points through which $S$ passes. 1984 Tokyo University entrance exam/Science

2005 Today's Calculation Of Integral, 16

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

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

2011 Today's Calculation Of Integral, 750

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

Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$