Mad scientist Kyouma writes $N$ positive integers on a board. Each second, he chooses two numbers $x, y$ written on the board with $x > y$, and writes the number $x^2-y^2$ on the board. After some time, he sends the list of all the numbers on the board to Christina. She notices that all the numbers from 1 to 1000 are present on the list. Aid Christina in finding the minimum possible value of N.

$PA_1A_2...A_n$ is a pyramid. The base $A_1A_2...A_n$ is a regular n-gon. The apex $P$ is placed so that the lines $PA_i$ all make an angle $60^{\cdot}$ with the plane of the base. For which $n$ is it possible to find $B_i$ on $PA_i$ for $i = 2, 3, ... , n$ such that $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$?

Prove that circles constructed on the sides of a convex quadrilateral as diameters completely cover this quadrilateral.

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

Find the number of functions $f : \{1, 2, . . . , n\} \to \{1995, 1996\}$ such that $f(1) + f(2) + ... + f(1996)$ is odd.

Ben and Connor are playing a game of wallball. The first player to lead by $2$ points wins the game. Suppose Ben wins each point with probability $\tfrac{3}{4}$ and is gracious enough to allow Connor to start with a $1$ point lead. The probability that Ben wins the game is $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ What is $m + n$?

we are given $n$ rectangles in the plane. Prove that between $4n$ right angles formed by these rectangles there are at least $[4\sqrt n]$ distinct right angles.

Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

On a board with $a$ rows and $b$ columns, each square has a switch and an unlit light bulb. By pressing the switch of a house, the lamp in that house changes state, along with the lamps in the same row and those in the same column (those that are on go out and the that are off light up). For what values of $a$ and $b$ is it possible to have just one lamp on, by pressing a series of switches?

Let A and B be two rectangles such that it is possible to get rectangle similar to A by putting together rectangles equal to B. Show that it is possible to get rectangle similar to B by putting together rectangles equal to A.

Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$ ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q \equal{} \overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.

Let $ r$ and $ n$ be nonnegative integers such that $ r \le n$. $ (a)$ Prove that: $ \frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}$ is an integer. $ (b)$ Prove that: $ \displaystyle\sum_{r\equal{}0}^{[n/2]}\frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}<2^{n\minus{}2}$ for all $ n \ge 9$.

Let $ABC$ be an acute-angled triangle, with $AC \neq BC$ and let $O$ be its circumcenter. Let $P$ and $Q$ be points such that $BOAP$ and $COPQ$ are parallelograms. Show that $Q$ is the orthocenter of $ABC$.

An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. [i]Proposed by Hojoo Lee, Korea[/i]

Show that for every integer $n \ge 3$ there exists positive integers $x_1, x_2, . . . , x_n$, pairwise different, so that $\{2, n\} \subset \{x_1, x_2, . . . , x_n\}$ and $$\frac{1}{x_1}+\frac{1}{x_2}+.. +\frac{1}{x_n}= 1.$$

Seven triangles of area $7$ lie in a square of area $27$. Prove that among the $7$ triangles there are $2$ that intersect in a region of area not less than $1$.

Find the largest $ 85$-digit integer which has property: the sum of its digits equals to the product of its digits.

Prove that if $n$ is a positive integer such that the equation \[x^{3}-3xy^{2}+y^{3}=n.\] has a solution in integers $(x,y),$ then it has at least three such solutions. Show that the equation has no solutions in integers when $n=2891$.

Let triangle $ABC$ be an acute triangle with $AB\neq AC$. The bisector of $BC$ intersects the lines $AB$ and $AC$ at points $F$ and $E$, respectively. The circumcircle of triangle $AEF$ has center $P$ and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$. Prove that the line $PD$ is tangent to the circumcircle of triangle $ABC$.

Let $N_0$ denote the set of nonnegative integers and $Z$ the set of all integers. Let a function $f : N_0 \times Z \to Z$ satisfy the conditions (i) $f(0, 0) = 1$, $f(0, 1) = 1$ (ii) for all $k, k \ne 0, k \ne 1$, $f(0, k) = 0$ and (iii) for all $n \ge 1$ and $k, f(n, k) = f(n -1, k) + f(n- 1, k - 2n)$. Find the value of $$\sum_{k=0}^{2009 \choose 2} f(2008, k)$$

A pizza is divided into six slices. Each slice contains one olive. One plays the following game. At each move it is allowed to move an olive on a neighboring slice. Is it possible to bring all the olives on one slice by exactly $20$ moves?

Do integers $a, b$ exist such that $a^{2006} + b^{2006} + 1$ is divisible by $2006^2$?

For any $a$, $b$, $c > 0$ and for any $n \in \mathbb{N}^*$, prove the inequality$$(a - b)\left(\frac{a}{b}\right)^n+(b - c)\left(\frac{b}{c}\right)^n+(c - a)\left(\frac{c}{a}\right)^n\geq (a - b)\frac{a}{b}+(b - c)\frac{b}{c}+(c - a)\frac{c}{a}$$

Let $P$ and $Q$ be points on a circle $k$. A chord $AC$ of $k$ passes through the midpoint $M$ of $PQ$. Consider a trapezoid $ABCD$ inscribed in $k$ with $AB \parallel PQ \parallel CD$. Prove that the intersection point $X$ of $AD$ and $BC$ depends only on $k$ and $P,Q.$

16 NHL teams in the first playoff round divided in pairs and to play series until 4 wins (thus the series could finish with score 4-0, 4-1, 4-2, or 4-3). After that 8 winners of the series play the second playoff round divided into 4 pairs to play series until 4 wins, and so on. After all the final round is over, it happens that $k$ teams have non-negative balance of wins (for example, the team that won in the first round with a score of 4-2 and lost in the second with a score of 4-3 fits the condition: it has $4+3=7$ wins and $2+4=6$ losses). Find the least possible $k$.