Olympiad problems

2007 Today's Calculation Of Integral, 237

Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$

2012 Today's Calculation Of Integral, 773

Tags: calculus , integration , calculus computations , logarithm

For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.

2011 Today's Calculation Of Integral, 680

Tags: calculus , integration , calculus computations

Let $a>0$. Evaluate $\int_0^a x^2\left(1-\frac{x}{a}\right)^adx$. [i]2011 Keio University entrance exam/Science and Technology[/i]

2007 Moldova National Olympiad, 12.7

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

Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

1991 Arnold's Trivium, 1

Tags: calculus , derivative , integration , function , analytic geometry , graphing lines , slope

Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.

2009 German National Olympiad, 1

Tags: geometry , 3d geometry , function , calculus , algebra

Find all non-negative real numbers $a$ such that the equation \[ \sqrt[3]{1+x}+\sqrt[3]{1-x}=a \] has at least one real solution $x$ with $0 \leq x \leq 1$. For all such $a$, what is $x$?

2008 Iran MO (3rd Round), 1

Tags: algebra , polynomial , trigonometry , function , calculus , integration , ratio

Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) \equal{} az^{2n \plus{} 1} \plus{} bz^{2n} \plus{} \bar bz \plus{} \bar a$ has a root on unit circle

Today's calculation of integrals, 850

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\pi} \{(1-x\sin 2x)e^{\cos ^2 x}+(1+x\sin 2x)e^{\sin ^ 2 x}\}\ dx.\]

2004 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , geometry

A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at $1$ meter per second. Suddenly someone turns on the turntable; it spins at $30$ rpm. Consider the set $S$ of points the mouse can reach in his car within $1$ second after the turntable is set in motion. What is the area of $S$, in square meters?

2002 AMC 12/AHSME, 24

Tags: trigonometry , geometry , calculus , integration , modular arithmetic

Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$. $ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$

1995 Brazil National Olympiad, 5

Tags: algebra , polynomial , calculus , integration , group theory , abstract algebra , galois theory

Show that no one $n$-th root of a rational (for $n$ a positive integer) can be a root of the polynomial $x^5 - x^4 - 4x^3 + 4x^2 + 2$.

2011 Today's Calculation Of Integral, 688

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

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

1995 IberoAmerican, 3

Tags: function , floor function , induction , calculus , integration , algebra

A function $f: \N\rightarrow\N$ is circular if for every $p\in\N$ there exists $n\in\N,\ n\leq{p}$ such that $f^n(p)=p$ ($f$ composed with itself $n$ times) The function $f$ has repulsion degree $k>0$ if for every $p\in\N$ $f^i(p)\neq{p}$ for every $i=1,2,\dots,\lfloor{kp}\rfloor$. Determine the maximum repulsion degree can have a circular function. [b]Note:[/b] Here $\lfloor{x}\rfloor$ is the integer part of $x$.

2011 Putnam, B5

Tags: integration , function , inequalities , calculus , vector

Let $a_1,a_2,\dots$ be real numbers. Suppose there is a constant $A$ such that for all $n,$ \[\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.\] Prove there is a constant $B>0$ such that for all $n,$ \[\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.\]

2009 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , function , analytic geometry , graphing lines , slope

Let $f$ be a differentiable real-valued function defined on the positive real numbers. The tangent lines to the graph of $f$ always meet the $y$-axis 1 unit lower than where they meet the function. If $f(1)=0$, what is $f(2)$?

2010 Today's Calculation Of Integral, 668

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

Consider two curves $y=\sin x,\ y=\sin 2x$ in $0\leq x\leq 2\pi$. (1) Let $(\alpha ,\ \beta)\ (0<\alpha <\pi)$ be the intersection point of the curves. If $\sin x-\sin 2x$ has a local minimum at $x=x_1$ and a local maximum at $x=x_2$, then find the values of $\cos x_1,\ \cos x_1\cos x_2$. (2) Find the area enclosed by the curves, then find the volume of the part generated by a rotation of the part of $\alpha \leq x\leq \pi$ for the figure about the line $y=-1$. [i]2011 Kyorin　University entrance exam/Medicine [/i]

1991 Arnold's Trivium, 52

Tags: calculus , integration , function

Calculate the first term of the asymptotic expression as $k\to\infty$ of the integral \[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{\sqrt{1+x^{2n}}}dx\]

2010 Today's Calculation Of Integral, 524

Tags: calculus , integration , function , calculus computations

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

Tags: calculus , integration , limit , logarithm , real analysis

For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that : (i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$ (ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$

2003 IMC, 2

Tags: calculus , integration , limit , trigonometry , inequalities , real analysis , logarithm

Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)

2012 Today's Calculation Of Integral, 833

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

Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$ For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$

2008 AIME Problems, 13

Tags: geometry , geometric transformation , reflection , calculus , trigonometry , integration , analytic geometry

A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $ R$ be the region outside the hexagon, and let $ S\equal{}\{\frac{1}{z}|z\in R\}$. Then the area of $ S$ has the form $ a\pi\plus{}\sqrt{b}$, where $ a$ and $ b$ are positive integers. Find $ a\plus{}b$.

2005 Putnam, B5

Tags: algebra , polynomial , calculus , derivative , system of equations

Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that (a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically) and that (b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$ Show that $P=0$ identically.

2010 Today's Calculation Of Integral, 567

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

Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\sin x\ (0\leq x\leq \pi)$ and the $x$-axis and denote $T$ by the area of the figure bounded by the curves $y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\ y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$ and the $x$-axis. Find the value of $a$ such that $ S: T=3: 1$.

2007 Today's Calculation Of Integral, 210

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

Evaluate $\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx$.