A [i]peacock [/i] is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock.

Given a circle of radius 2, there are many line segments of length 2 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. $\text{(A)}\ \frac \pi 4 \qquad \text{(B)}\ 4 - \pi \qquad \text{(C)}\ \frac \pi 2 \qquad \text{(D)}\ \pi \qquad \text{(E)}\ 2\pi$

You are given a balance and one copy of each of ten weights of $1, 2, 4, 8, 16, 32, 64, 128, 256$ and $512$ grams. An object weighing $M$ grams, where $M$ is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance. (a) Prove that no object may be balanced in more than $89$ ways. (b) Find a value of $M$ such that an object weighing $M$ grams can be balanced in $89$ ways. (A Shapovalov, A Kulakov)

In $1991$ the population of a town was a perfect square. Ten years later, after an increase of $150$ people, the population was $9$ more than a perfect square. Now, in $2011$, with an increase of another $150$ people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period? $ \textbf{(A)}\ 42 \qquad\textbf{(B)}\ 47 \qquad\textbf{(C)}\ 52\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 62 $

Let $ABCD$ be a tetrahedron and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that for every point $P \in (MN)$ with $P \neq M$ and $P \neq N$, there exist unique points $X$ and $Y$ on segments $AB$ and $CD$, respectively, such that $X,P,Y$ are collinear.

Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins. $(a)$ Determine which player has a winning strategy for $N=2005$. $(b)$ Determine which player has a winning strategy for $N=2004$. $(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.

Let $S$ be a finite subset of a straight line. Say that $S$ has the [i]repeated distance property [/i] if every value of the distance between two points of $S$ (except the longest) occurs at least twice. Show that if $S$ has the [i]repeated distance property [/i] then the ratio of any two distances between two points of $S$ is rational.

Prove that for all positive integers $a, b, c$ the following inequality holds: $$\frac{a + b}{a + c}+\frac{b + c}{b + a}+\frac{c + a}{c + b} \le \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$

A $9 \times 10$ board is divided into $90$ unit cells. There are certain rules for moving a non-standard chess queen from one square to another: [list] [*]The queen can only move along the column or row it is in each step. [*]For any natural number $n$, if $x$ cells move made in $(2n-1)$th step, then $9-x$ cells move will be done in $(2n)$th step. The last cell it stops at during these steps is considered the visited cell. [/list] Is it possible for the queen to move from any square on the board and return to the square where it started after visiting all the squares of the board exactly once? Note: At each step, the queen chooses the right, left, up, and down direction within the above condition can choose.

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $ 32$? $ \textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255$

Denote by $a_n$ the last digit of the number $n^{(n^n)}$ (let $n\ne 0$ be a natural number ). Prove that the numbers $a_n$ form a periodic sequence and state this period!

How many hours does it take a train traveling at an average rate of $ 40$ mph between stops to travel $ a$ miles it makes $ n$ stops of $ m$ minutes each? $ \textbf{(A)}\ \frac{3a\plus{}2mn}{120} \qquad \textbf{(B)}\ 3a\plus{}2mn \qquad \textbf{(C)}\ \frac{3a\plus{}2mn}{12} \qquad \textbf{(D)}\ \frac{a\plus{}mn}{40} \qquad \textbf{(E)}\ \frac{a\plus{}40mn}{40}$

Points $K,L,M$ on the sides $AB,BC,CA$ respectively of a triangle $ABC$ satisfy $\frac{AK}{KB} = \frac{BL}{LC} = \frac{CM}{MA}$. Show that the triangles $ABC$ and $KLM$ have a common orthocenter if and only if $\triangle ABC$ is equilateral.

Does there exist a piecewise linear function $f$ defined on the segment [$-1,1]$ (including the ends) such that $f(f(x)) = -x$ for all x? (A function is called piecewise linear if its graph is the union of a finite set of points and intervals; it may be discontinuous).

$A, B, C$, and $D$ are the four different points on the circle $O$ in the order. Let the centre of the scribed circle of triangle $ABC$, which is tangent to $BC$, be $O_1$. Let the centre of the scribed circle of triangle $ACD$, which is tangent to $CD$, be $O_2$. (1) Show that the circumcentre of triangle $ABO_1$ is on the circle $O$. (2) Show that the circumcircle of triangle $CO_1O_2$ always pass through a fixed point on the circle $O$, when $C$ is moving along arc $BD$.

The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis? $\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\frac{1}{2}\right)$

A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $ \frac{3}{4}$ of its former rate and arrives $ 3 \frac{1}{2}$ hours late. Had the accident happened $ 90$ miles farther along the line, it would have arrived only $ 3$ hours late. The length of the trip in miles was: $ \textbf{(A)}\ 400 \qquad \textbf{(B)}\ 465 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 640 \qquad \textbf{(E)}\ 550$

Two players are writting in turn natural numbers not exceeding $p$. The rules forbid to write the divisors of the numbers already having been written. Those who cannot make his move looses. a) Who, and how, can win if $p=10$? b) Who wins if $p=1000$?

Ivan and Kolya play a game, Ivan starts. Initially, the polynomial $x-1$ is written of the blackboard. On one move, the player deletes the current polynomial $f(x)$ and replaces it with $ax^{n+1}-f(-x)-2$, where $\deg(f)=n$ and $a$ is a real root of $f$. The player who writes a polynomial which does not have real roots loses. Can Ivan beat Kolya?

We consider the real numbers $a _ 1, a _ 2, a _ 3, a _ 4, a _ 5$ with the zero sum and the property that $| a _ i - a _ j | \le 1$ , whatever it may be $i,j \in \{1, 2, 3, 4, 5 \} $. Show that $a _ 1 ^ 2 + a _ 2 ^ 2 + a _ 3 ^ 2 + a _ 4 ^ 2 + a _ 5 ^ 2 \le \frac {6} {5}$ .

Let $p > 5$ be a prime number. Show that there exists a prime number $q < p$ and a positive integer $n$ such that $p$ divides $n^2-q$. [i]Proposed by Andrew Gu.[/i]

Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place: [color=#0000FF]Alice:[/color] Are you going to cover your keys? [color=#FF0000]Bob:[/color] I would like to, but there are only $7$ colours and I have $8$ keys. [color=#0000FF]Alice:[/color] Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key. [color=#FF0000]Bob:[/color] You must be careful what you mean by "[i]next to[/i]" or "[i]three keys over from[/i]" since you can turn the key ring over and the keys are arranged in a circle. [color=#0000FF]Alice:[/color] Even so, you don't need $8$ colours. [b]Problem:[/b] What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered.

A person enters the social network facebook. He befriends at least one person a day for the first $30$ days. At the end of those $30$ days, it has been exactly $45$ friends. Prove that there is a sequence of consecutive days where made exactly $14$ friends.

Let $E$ be the ellipse lying in the $x, y$ plane centered at $(0, 0)$ with semi-major axis of length $2$ along the $x$-axis and semi-minor axis of length $1$ along the $y$-axis. Let $C$ be a cone created by revolving two perpendicular lines about an angle bisector of the perpendicular angle. There are some points $(x, y, z)$ where the vertex of $C$ could be so that $E$ is the intersection of $C$ with the $x, y$ plane. These points define a convex polygon in the $x, z$ plane. The area of this polygon can be expressed as $\sqrt{n}$ for some positive integer $n.$ Find $n.$ (Some definitions: the semi-major axis is the longest distance from the center of the ellipse to the boundary, and the semi-minor axis is the shortest distance from the center of the ellipse to the boundary.)