Olympiad problems

2010 Today's Calculation Of Integral, 630

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

2010 Today's Calculation Of Integral, 624

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

Find the continuous function $f(x)$ such that the following equation holds for any real number $x$. \[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\] [i]1977 Keio University entrance exam/Medicine[/i]

2012 Today's Calculation Of Integral, 848

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

Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$

2012 Today's Calculation Of Integral, 858

Tags: calculus , integration , perimeter , calculus computations , geometry

On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$. $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$. $(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.

2007 Today's Calculation Of Integral, 224

Tags: calculus , integration , trigonometry , calculus computations

Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

2012 Today's Calculation Of Integral, 823

Tags: calculus , integration , trigonometry , geometry , geometric transformation , rotation , calculus computations

Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$ (1) Express $y$ in terms of $x$. (2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$. (3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.

2012 Today's Calculation Of Integral, 856

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

On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

2007 Today's Calculation Of Integral, 216

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

Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$. Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.

2011 Today's Calculation Of Integral, 691

Tags: calculus , integration , geometry , geometric transformation , rotation , calculus computations , logarithm

Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$. Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis. [i]2011 Yokohama National Universty entrance exam/Engineering[/i]

2013 Waseda University Entrance Examination, 3

Tags: function , calculus , calculus computations , logarithm

Let $f(x)=\frac 12e^{2x}+2e^x+x$. Answer the following questions. (1) For a real number $t$, set $g(x)=tx-f(x).$ When $x$ moves in the range of all real numbers, find the range of $t$ for which $g(x)$ has maximum value, then for the range of $t$, find the maximum value of $g(x)$ and the value of $x$ which gives the maximum value. (2) Denote by $m(t)$ the maximum value found in $(1)$. Let $a$ be a constant, consider a function of $t$, $h(t)=at-m(t)$. When $t$ moves in the range of $t$ found in $(1)$, find the maximum value of $h(t)$.

2005 Today's Calculation Of Integral, 58

Tags: calculus , integration , calculus computations

Let $f(x)=\frac{e^x}{e^x+1}$ Prove the following equation. \[\int_a^b f(x)dx+\int_{f(a)}^{f(b)} f^{-1}(x)dx=bf(b)-af(a)\]

Today's calculation of integrals, 849

Tags: calculus , integration , function , calculus computations

Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

2010 Today's Calculation Of Integral, 649

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

Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$, $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ [i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]

2006 ISI B.Math Entrance Exam, 4

Tags: calculus , derivative , function , induction , calculus computations

Let $f:\mathbb{R} \to \mathbb{R}$ be a function that is a function that is differentiable $n+1$ times for some positive integer $n$ . The $i^{th}$ derivative of $f$ is denoted by $f^{(i)}$ . Suppose- $f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0$. Prove that $f^{(n+1)}(x)=0$ for some $x \in (0,1)$

2005 Today's Calculation Of Integral, 77

Tags: calculus , integration , geometry , calculus computations

Find the area of the part enclosed by the following curve. \[x^2+2axy+y^2=1\ (-1<a<1)\]

2007 Today's Calculation Of Integral, 171

Tags: calculus , integration , calculus computations

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

2009 Today's Calculation Of Integral, 467

Tags: calculus , integration , geometric transformation , rotation , trigonometry , calculus computations , geometry

Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$. (1) Find the maximum value of $ y$. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis. (3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.

2007 Today's Calculation Of Integral, 197

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

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

2005 Today's Calculation Of Integral, 13

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

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

2005 Today's Calculation Of Integral, 60

Tags: calculus , integration , trigonometry , calculus computations

Let $a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)$ Evaluate \[\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})\]

2009 Today's Calculation Of Integral, 519

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx$.

2007 Moldova National Olympiad, 12.4

Tags: calculus , integration , inequalities , function , calculus computations

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]

Today's calculation of integrals, 900

Tags: calculus , integration , trigonometry , calculus computations

Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$

2007 Today's Calculation Of Integral, 169

Tags: calculus , integration , function , calculus computations

(1) Let $f(x)$ be the differentiable and increasing function such that $f(0)=0.$Prove that $\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}.$ (2) $g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots)$ satisfies $\int_{-1}^{1}(px+q)g_{n}(x)dx=0$ for all linear equations $px+q.$ Find $a_{n},\ b_{n}.$

2010 Today's Calculation Of Integral, 638

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

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