Olympiad problems

2006 Iran MO (3rd Round), 3

Tags: invariant , number theory , integration , calculus , abstract algebra , group theory

$L$ is a fullrank lattice in $\mathbb R^{2}$ and $K$ is a sub-lattice of $L$, that $\frac{A(K)}{A(L)}=m$. If $m$ is the least number that for each $x\in L$, $mx$ is in $K$. Prove that there exists a basis $\{x_{1},x_{2}\}$ for $L$ that $\{x_{1},mx_{2}\}$ is a basis for $K$.

1993 AMC 12/AHSME, 30

Tags: calculus , integration , function

Given $0 \le x_0 <1$, let \[ x_n= \begin{cases} 2x_{n-1} & \text{if}\ 2x_{n-1} <1 \\ 2x_{n-1}-1 & \text{if}\ 2x_{n-1} \ge 1 \end{cases} \] for all integers $n>0$. For how many $x_0$ is it true that $x_0=x_5$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ \text{infinitely many} $

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4

Tags: calculus , derivative , topology

Define mapping $F : \mathbb{R}^4\rightarrow \mathbb{R}^4$ as $F(x,\ y,\ z,\ w)=(xy,\ y,\ z,\ w)$ and let mapping $f : S^3\rightarrow \mathbb{R}^4$ be restriction of $F$ to 3 dimensional ball $S^3=\{(x,\ y,\ z,\ w)\in{\mathbb{R}^4} | x^2+y^2+z^2+w^2=1\}$. Find the rank of $df_p$, or the differentiation of $f$ at every point $p$ in $S^3$.

1973 AMC 12/AHSME, 16

Tags: integration , calculus , algebra , quadratic , quadratic formula

If the sum of all the angles except one of a convex polygon is $ 2190^{\circ}$, then the number of sides of the polygon must be $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 21$

Today's calculation of integrals, 866

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

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 $x=\sin t.$

1997 IberoAmerican, 1

Tags: floor function , integration , number theory , calculus , ratio

Let $r\geq1$ be areal number that holds with the property that for each pair of positive integer numbers $m$ and $n$, with $n$ a multiple of $m$, it is true that $\lfloor{nr}\rfloor$ is multiple of $\lfloor{mr}\rfloor$. Show that $r$ has to be an integer number. [b]Note: [/b][i]If $x$ is a real number, $\lfloor{x}\rfloor$ is the greatest integer lower than or equal to $x$}.[/i]

2000 CentroAmerican, 1

Tags: integration , calculus

Find all three-digit numbers $ abc$ (with $ a \neq 0$) such that $ a^{2}+b^{2}+c^{2}$ is a divisor of 26.

1969 Miklós Schweitzer, 5

Tags: real analysis , integration , function , calculus

Find all continuous real functions $ f,g$ and $ h$ defined on the set of positive real numbers and satisfying the relation \[ f(x\plus{}y)\plus{}g(xy)\equal{}h(x)\plus{}h(y)\] for all $ x>0$ and $ y>0$. [i]Z. Daroczy[/i]

2009 ISI B.Math Entrance Exam, 2

Tags: algebra , equation , calculus

Let $c$ be a fixed real number. Show that a root of the equation \[x(x+1)(x+2)\cdots(x+2009)=c\] can have multiplicity at most $2$. Determine the number of values of $c$ for which the equation has a root of multiplicity $2$.

2005 Today's Calculation Of Integral, 2

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

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

1997 VJIMC, Problem 3

Tags: integration , inequalities , calculus , multivariable calculus

Let $u\in C^2(\overline D)$, $u=0$ on $\partial D$ where $D$ is the open unit ball in $\mathbb R^3$. Prove that the following inequality holds for all $\varepsilon>0$: $$\int_D|\nabla u|^2dV\le\varepsilon\int_D(\Delta u)^2dV+\frac1{4\varepsilon}\int_Du^2dV.$$(We recall that $\nabla u$ and $\Delta u$ are the gradient and Laplacian, respectively.)

2005 Today's Calculation Of Integral, 56

Tags: limit , integration , calculus computations , calculus

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

Today's calculation of integrals, 890

Tags: algorithm , integration , function , calculus , calculus computations

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

2007 Today's Calculation Of Integral, 203

Tags: integration , trigonometry , calculus computations , calculus

Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate the following definite integral. \[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]

2011 Today's Calculation Of Integral, 747

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

Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$

2010 Today's Calculation Of Integral, 667

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

Let $a>1,\ 0\leq x\leq \frac{\pi}{4}$. Find the volume of the solid generated by a rotation of the part bounded by two curves $y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}$ about the $x$-axis. [i]1993 Hiroshima Un iversity entrance exam/Science[/i]

2004 Harvard-MIT Mathematics Tournament, 2

Tags: function , calculus , derivative

Suppose the function $f(x)-f(2x)$ has derivative $5$ at $x=1$ and derivative $7$ at $x=2$. Find the derivative of $f(x)-f(4x)$ at $x=1$.

2008 ISI B.Math Entrance Exam, 1

Tags: function , integration , calculus , calculus computations

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function . Suppose \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$ and all $t>0$ . Then show that there exists a constant $c$ such that $f(x)=cx\ \forall x$

2012 ISI Entrance Examination, 2

Tags: limit , derivative , calculus , calculus computations , function

Consider the following function \[g(x)=(\alpha+|x|)^{2}e^{(5-|x|)^{2}}\] [b]i)[/b] Find all the values of $\alpha$ for which $g(x)$ is continuous for all $x\in\mathbb{R}$ [b]ii)[/b]Find all the values of $\alpha$ for which $g(x)$ is differentiable for all $x\in\mathbb{R}$.

1991 Arnold's Trivium, 11

Tags: integration , matrix , symmetry , linear algebra , calculus

Investigate the convergence of the integral \[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dxdy}{1+x^4y^4}\]

2013 F = Ma, 16

Tags: derivative , calculus

A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way. How much time passes until the cloud collapses fully? (The constant $0.5427$ is actually $\sqrt{\frac{3 \pi}{32}}$.) $\textbf{(A) } \frac{0.5427}{r_0^2 \sqrt{G \rho_0}}\\ \\ \textbf{(B) } \frac{0.5427}{r_0 \sqrt{G \rho_0}}\\ \\ \textbf{(C) } \frac{0.5427}{\sqrt{r_0} \sqrt{G \rho_0}}\\ \\ \textbf{(D) } \frac{0.5427}{\sqrt{G \rho_0}}\\ \\ \textbf{(E) } \frac{0.5427}{\sqrt{G \rho_0}}r_0$

Today's calculation of integrals, 870

Tags: calculus , inequalities , integration , conic , ellipse , hyperbola , geometry

Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]