On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$.

3.Let ABCD be a parallelogram with side AB longer than AD and acute angle $\angle DAB$. The bisector of ∠DAB meets side CD at L and line BC at K. If O is the circumcenter of triangle LCK, prove that the points B,C,O,D lie on a circle.

[b]Problem 2[/b] Consider the infinite chessboard whose rows and columns are indexed by positive integers. Is it possible to put a single positive rational number into each cell of the chessboard so that each positive rational number appears exactly once and the sum of every row and of every column is finite?

Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$

Let $a_{1},a_{2},...,a_{1999}$ be a sequence of nonnegative integers such that for any $i,j$ with $i+j\leq 1999$ , $a_{i}+a_{j}\leq a_{i+j}\leq a_{i}+a_{j}+1$. Prove that there exists a real number $x$ such that $a_{n}=[nx]\forall n$.

Find integer solutions of the equation $8x^3 - 4 = y(6x - y^2)$

suppose that $G<S_n$ is a subgroup of permutations of $\{1,...,n\}$ with this property that for every $e\neq g\in G$ there exist exactly one $k\in \{1,...,n\}$ such that $g.k=k$. prove that there exist one $k\in \{1,...,n\}$ such that for every $g\in G$ we have $g.k=k$.(20 points)

For a positive integer $n$, if $n$ is a product of two different primes and $n \equiv 2 \pmod 3$, then $n$ is called "special number." For example, $14, 26, 35, 38$ is only special numbers among positive integers $1$ to $50$. Prove that for any finite set $S$ with special numbers, there exist two sets $A, B$ such that [list] [*] $A \cap B = \emptyset, A \cup B = S$ [*] $||A| - |B|| \leq 1$ [*] For all primes $p$, the difference between number of elements in $A$ which is multiple of $p$ and number of elements in $B$ which is multiple of $p$ is less than or equal to $1$. [/list]

prove that for almost every real number $\alpha \in [0,1]$ there exists natural number $n_{\alpha} \in \mathbb N$ such that the inequality $|\alpha-\frac{p}{q}|\le \frac{1}{q^n}$ for natural $n\ge n_{\alpha}$ and rational $\frac{p}{q}$ has no answers.

Let $B$ be a $1 \times 2 \times 4$ box (rectangular parallelepiped). Let $R$ be the set of points that are within distance 3 of some point in $B$. (Note that $R$ contains $B$.) What is the volume of $R$?

Let $\mathcal{P}$ be the set of all polynomials $p(x)=x^4+2x^2+mx+n$, where $m$ and $n$ range over the positive reals. There exists a unique $p(x) \in \mathcal{P}$ such that $p(x)$ has a real root, $m$ is minimized, and $p(1)=99$. Find $n$. [i]Proposed by [b]AOPS12142015[/b][/i]

a) Two seventh graders and several eightth graders take part in a chess tournament. The two seventh graders together scored eight points. The scores of eightth graders are equal. How many eightth graders took part in the tournament? b) Ninth and tenth graders participated in a chess tournament. There were ten times as many tenth graders as ninth graders. The total score of tenth graders was $4.5$ times that of the ninth graders. What was the ninth graders score? Note: According to the rules of a chess tournament, each of the tournament participants ra plays one game with each of them. If one of the players wins the game, then he gets one point, and his opponent gets zero points. In case of a tie, the players receive 1/2 point.

In Tlaxcala, there is a transportation system that works through buses that travel from one city to another in one direction . A set $S$ of cities is said [i]beautiful[/i] if it contains at least three different cities and from each city $A$ in $S$ at least two buses depart, each one goes directly to a different city in $S$ and none of them is $A$ (if there is a direct bus from $A$ to a city $B$ in $S$, there is not necessarily a direct bus from $B$ to $A$). Show that if there exists a beautiful set of cities $S$, then there exists a beautiful $T$ subset of $S$, such that for any two cities in $T$, you can get from one to another by taking buses that only pass through cities in $T$. Note: A bus goes directly from one city to another if it does not pass through any other city.

Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of $n$ between $1$ and $2013$ (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes. [i]Ray Li[/i]

Find all prime numbers that do not have a multiple ending in $2015$.

(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$? (b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positive divisors of $ n$?

Let $p$ a prime number, $p\geq 5$. Find the number of polynomials of the form \[ x^p + px^k + p x^l + 1, \quad k > l, \quad k, l \in \left\{1,2,\dots,p-1\right\}, \] which are irreducible in $\mathbb{Z}[X]$. [i]Valentin Vornicu[/i]

Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation \[n^3+m^3-nm(n+m)=2023.\]

A function $f$ defined on integers such that $f (n) =n + 3$ if $n$ is odd $f (n) = \frac{n}{2}$ if $n$ is even If $k$ is an odd integer, determine the values for which $f (f (f (k))) = k$.

Four congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four congruent spheres so that all four are internally tangent to $R$. A smaller sphere, $S$, sits in the space between the four congruent spheres so that all four are externally tangent to $S$. The ratio of the surface area of $R$ to the surface area of $S$ can be written $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.

Prove that an arbitrary triangle with area $1$ can be covered by an isosceles triangle with area less than $\sqrt{2}$.

$\boxed{\text{G1}}$ Let $M$ be interior point of the triangle $ABC$ with <BAC=70and <ABC=80 If <ACM=10 and <CBM=20.Prove that $AB=MC$

A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$. Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.

Into each box of a $ 2012 \times 2012 $ square grid, a real number greater than or equal to $ 0 $ and less than or equal to $ 1 $ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to $ 1 $, no matter how the grid is split into $2$ such rectangles. Determine the maximum possible value for the sum of all the $ 2012 \times 2012 $ numbers inserted into the boxes.

In $\triangle ABC$, let $X$ and $Y$ be points on segment $BC$ such that $AX=XB=20$ and $AY=YC=21$. Let $J$ be the $A$-excenter of triangle $\triangle AXY$. Given that $J$ lies on the circumcircle of $\triangle ABC$, the length of $BC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Andrew Wen[/i]