An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors.

A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$. a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?

Let $f(x)=a_0+a_1|x|+a_2|x|^2+a_3|x|^3$, where $a_0,a_1,a_2,a_3$ are constant. Then $\textbf{(A)}~f(x)\text{ is differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$ $\textbf{(B)}~f(x)\text{ is not differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$ $\textbf{(C)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0$ $\textbf{(D)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0,a_3=0$

Given a set $A$ and a function $f: A\rightarrow A$, denote by $f_{1}(A)=f(A)$, $f_{2}(A)=f(f_{1}(A))$, $f_{3}(A)=f(f_{2}(A))$, and so on, ($f_{n}(A)=f(f_{n-1}(A))$, where the notation $f(B)$ means the set $\{ f(x) \ : \ x\in B\}$ of images of points from $B$). Denote also by $f_{\infty}(A)=f_{1}(A)\cap f_{2}(A)\cap \ldots = \bigcap_{n\geq 1}f_{n}(A)$. a) Show that if $A$ is finite, then $f(f_{\infty}(A))=f_{\infty}(A)$. b) Determine if the above is true for $A=\mathbb{N}\times \mathbb{N}$ and the function \[f\big((m,n)\big)=\begin{cases}(m+1,n) & \mbox{if }n\geq m\geq 1 \\ (0,0) & \mbox{if }m>n \\ (0,n+1) & \mbox{if }n=0. \end{cases}\]

Let $ABC$ be an equilateral triangle of side length $2$. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k<1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$. with $CB'=AC'=k$. Line segments are drawn from points $A',B',C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Prove that $\Delta PQR$ is an equilateral triangle with side length ${4(1-k) \over \sqrt{k^2-2k+4}}$.

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

For every non-negative integer $i$, define the number $M(i)$ as follows: write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$. (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ ) (a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $. Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ . (b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ . Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$. (A Kanel)

Let $n$ be a positive integer. Ana and Banana play a game with $2n$ lamps numbered $1$ to $2n$ from left to right. Initially, all lamps numbered $1$ through $n$ are on, and all lamps numbered $n+1$ through $2n$ are off. They play with the following rules, where they alternate turns with Ana going first: [list] [*] On Ana's turn, she can choose two adjacent lamps $i$ and $i+1$, where lamp $i$ is on and lamp $i+1$ is off, and toggle both. [*] On Banana's turn, she can choose two adjacent lamps which are either both on or both off, and toggle both. [/list] Players must move on their turn if they are able to, and if at any point a player is not able to move on her turn, then the game ends. Determine all $n$ for which Banana can turn off all the lamps before the game ends, regardless of the moves that Ana makes. [i]Andrew Wen[/i]

All vertices of a regular 100-gon are colored in 10 colors. Prove that there exist 4 vertices of the given 100-gon which are the vertices of a rectangle and which are colored in at most 2 colors.

Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.

Determine the last digit of $5^5+6^6+7^7+8^8+9^9$.

The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b = \dfrac{a^2}{b}$. Determine $[(1\otimes 2)\otimes 3] - [1\otimes (2\otimes 3)]$. $\text{(A)}\ -\dfrac{2}{3} \qquad \text{(B)}\ -\dfrac{1}{4} \qquad \text{(C)}\ 0 \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{2}{3}$

Is it possible to represent number $2011... 2011$, where number $2011$ is written $20112011$ times, as a product of some number and sum of its digits?

Let $f(x,y)=(\cos x+y\sin x)^2$. We may express $\text{max}_xf(x,y)$, the maximum value of $f(x,y)$ over all values of $x$ for a given fixed value of $y$, as a function of $y$, call it $g(y)$. Let the smallest positive value $x$ which achieves this maximum value of $f(x,y)$ for a given $y$ be $h(y)$. Compute \[\int_1^{2+\sqrt{3}}\frac{h(y)}{g(y)}\text{d}y.\]

Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $M,N$. A line $\ell$ is tangent to $\Gamma_1 ,\Gamma_2$ at $A$ and $B$, respectively. The lines passing through $A$ and $B$ and perpendicular to $\ell$ intersects $MN$ at $C$ and $D$ respectively. Prove that $ABCD$ is a parallelogram.

a) In a triangle $ MNP$, the lenghts of the sides are less than $ 2$. Prove that the lenght of the altitude corresponding to the side $ MN$ is less than $ \sqrt {4 \minus{} \frac {MN^2}{4}}$. b) In a tetrahedron $ ABCD$, at least $ 5$ edges have their lenghts less than $ 2$.Prove that the volume of the tetrahedron is less than $ 1$.

The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.

Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions: $i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$. $ii)$ There are no two lines of $S$ which are parallel.

The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\). [i]Proposed by Karthik Vedula[/i]

Let be a sequence $ \left( b_n \right)_{n\ge 1} $ of integers, having the following properties: $ \text{(i)} $ the sequence $ \left( \frac{b_n}{n} \right)_{n\ge 1} $ is convergent. $ \text{(ii)} m-n|b_m-b_n, $ for any natural numbers $ m>n. $ Prove that there exists an index from which the sequence $ \left( b_n \right)_{n\ge 1} $ is an arithmetic one. [i]Cristinel Mortici[/i]

In rectangle $ABCD$, $AD=1$, $P$ is on $AB$ and $DB$ and $DP$ trisect $\angle ADC$. What is the perimeter $\triangle BDP$ [list=1] [*] $3+\frac{\sqrt{3}}{3}$ [*] $2+\frac{4\sqrt{3}}{3}$ [*] $2+2\sqrt{2}$ [*] $\frac{3+3\sqrt{5}}{2}$ [/list]

Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$. Prove that $ZA=ZH$. [i]Vadzim Kamianetski[/i]

The convex quadrilateral $ABCD$ has $|AB| = 8$, $|BC| = 29$, $|CD| = 24$ and $|DA| = 53$. What is the area of the quadrilateral if $\angle ABC + \angle BCD = 270^o$?