Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$. Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.

Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce? $(\textbf{A}) \: 24 \qquad (\textbf{B}) \: 30 \qquad (\textbf{C}) \: 48 \qquad (\textbf{D}) \: 60 \qquad (\textbf{E}) \: 64$

There are $n$ pawns on $n$ distinct squares of a $19\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely?

Let $Q_n$ be the product of the squares of even numbers less than or equal to $n$ and $K_n$ equal to the product of cubes of odd numbers less than or equal to $n$. What is the highest power of $98$, that [b]a)[/b]$Q_n$, [b]b)[/b] $K_n$ or [b]c)[/b] $Q_nK_n$ divides? If one divides $Q_{98}K_{98}$ by the highest power of $98$, then one get a number $N$. By which power-of-two number is $N$ still divisible?

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]