Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim.

Cross-shaped tiles are to be placed on a $8\times8$ square grid without overlapping. Find the largest possible number of tiles that can be placed. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy8zL2EyY2Q4MDcyMWZjM2FmZGFhODkxYTk5ZmFiMmMwNzk0MzZmYmVjLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wNyBhdCA2LjIzLjU4IEFNLnBuZw[/img]

A photographer took some pictures at a party with $10$ people. Each of the $45$ possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?

$n$ rooks and $k$ pawns are arranged on a $100 \times 100$ board. The rooks cannot leap over pawns. For which minimum $k$ is it possible that no rook can capture any other rook? Junior League: $n=2551$ ([i]Proposed by A. Kuznetsov[/i]) Senior League: $n=2550$ ([i]Proposed by N. Vlasova[/i])

Two pupils $A$ and $B$ play the following game. They begin with a pile of $1998$ matches and $A$ plays first. A player who is on turn must take a nonzero square number of matches from the pile. The winner is the one who makes the last move. Decide who has the winning strategy and give one such strategy.

Let $A_1, A_2, A_3$ be nonempty sets of integers such that for $\{i, j, k\} = \{1, 2, 3\}$ holds $$(x \in A_i, y\in A_j) \Rightarrow (x + y \in A_k, x - y \in A_k).$$ Prove that at least two of the sets $A_1, A_2, A_3$ are equal. Can any of these sets be disjoint?

Let the [i]subbishop[/i] (a bishop is the figure moving only by a diagonal) be a figure moving only by diagonal but only in the next cells (squares) of the chessboard. Find the maximal count of subbishops over a chessboard $n\times n$, no two of which are not attacking. [i]V. Chukanov[/i]

There are 100 saucers in a circle. Two people take turns putting marmalade of various colors in empty saucers. The first person can choose one or three empty saucers and fill each of them with marmalade of arbitrary color. The second one can choose one empty saucer and fill it with marmalade of arbitrary color. There should not be two adjacent saucers with marmalade of the same color. The game ends when all the saucers are filled. The loser is the last player to introduce a new color of marmalade into the game. Who has a winning strategy?

Given a succession $C$ of $1001$ positive real numbers (not necessarily distinct), and given a set $K$ of distinct positive integers, the permitted operation is: select a number $k\in{K}$, then select $k$ numbers in $C$, calculate the arithmetic mean of those $k$ numbers, and replace each of those $k$ selected numbers with the mean. If $K$ is a set such that for each $C$ we can reach, by a sequence of permitted operations, a state where all the numbers are equal, determine the smallest possible value of the maximum element of $K$.

Let $ n$ be a positive integer congruent to $1$ modulo $4$. Xantippa has a bag of $n + 1$ balls numbered from $ 0$ to $n$. She draws a ball (randomly, equally distributed) from the bag and reads its number: $k$, say. She keeps the ball and then picks up another $k$ balls from the bag (randomly, equally distributed, without repossession). Finally, she adds up the numbers of all the $k + 1$ balls she picked up. What is the probability that the sum will be odd?

There are two very strict laws in the country of Militaria. (i) Anyone who is shorter than $80\%$ (or more) of his “neighbours” (i.e. men living at distance less then $r$ from him) is freed from the military service. (ii) Anyone who is taller than $80\%$ (or more) of his “neighbours” (i.e. men living at distance less then $R$ from him) is allowed to serve in the police. A nice thing is that each man $X$ may choose his own (possibly different) positive numbers $r = r(X)$ and $R = R(X)$. Can it happen that $90\%$ (or more) of the men in Militaria are free from the army and, at the same time, $90\%$ (or more) of the men in Militaria are allowed to serve in the police? (The places of living of the men are fixed points in the plane.) (N Konstantinov)

Let $M$ be a set of $2017$ positive integers. For any subset $A$ of $M$ we define $f(A) := \{x\in M\mid \text{ the number of the members of }A\,,\, x \text{ is multiple of, is odd }\}$. Find the minimal natural number $k$, satisfying the condition: for any $M$, we can color all the subsets of $M$ with $k$ colors, such that whenever $A\neq f(A)$, $A$ and $f(A)$ are colored with different colors.

Find all $(m,n) \in \mathbb{Z}^2$ that we can color each unit square of $m \times n$ with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.

Let $\pi$ be a randomly chosen permutation of the numbers from $1$ through $2012$. Find the probability that$$ \pi (\pi(2012)) = 2012.$$

Let $n\geq 2$ be a positive integer. On an $n\times n$ board, $n$ rooks are placed in such a manner that no two attack each other. All rooks move at the same time and are only allowed to move in a square adjacent to the one in which they are located. Determine all the values ​​of $n$ for which there is a placement of the rooks so that, after a move, the rooks still do not attack each other. [i]Note: Two squares are adjacent if they share a common side.[/i]

$(GDR 3)$ Find the number of permutations $a_1, \cdots, a_n$ of the set $\{1, 2, . . ., n\}$ such that $|a_i - a_{i+1}| \neq 1$ for all $i = 1, 2, . . ., n - 1.$ Find a recurrence formula and evaluate the number of such permutations for $n \le 6.$

Show that the number of non-zero integers in the expansion of the $n$-th order determinant having zeroes in the main diagonal and ones elsewhere is $$n ! \left(1- \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^{n}}{n!} \right) .$$

Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of $n$.

$ 2n\plus{}1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?

On a circle, there are $1000$ nonzero real numbers painted black and white in turn. Each black number is equal to the sum of two white numbers adjacent to it, and each white number is equal to the product of two black numbers adjacent to it. What are the possible values of the total sum of $1000$ numbers?

In country there are some cities, some pairs of cities are connected with roads.From every city go out exactly $100$ roads. We call $10$ roads, that go out from same city, as bunch. Prove, that we can split all roads in several bunches.

Two players, $A$ and $B$, play a game on a board which is a rhombus of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles, as shown in the diagram for $n=4$. $A$ uses a red token and $B$ uses a blue token, which are initially placed in cells containing opposite corners of the board (the $60^{\circ}$ ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started. If $A$ starts the game, determine which player has a winning strategy.

Let $n$ be the number of ordered $5$-tuples $(a_1,a_2,\ldots,a_5)$ of positive integers such that $\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_5}=1$. Is $n$ an even number?

Before The World Cup tournament, the football coach of $F$ country will let seven players, $A_1, A_2, \ldots, A_7$, join three training matches (90 minutes each) in order to assess them. Suppose, at any moment during a match, one and only one of them enters the field, and the total time (which is measured in minutes) on the field for each one of $A_1, A_2, A_3$, and $A_4$ is divisible by $7$ and the total time for each of $A_5, A_6$, and $A_7$ is divisible by $13$. If there is no restriction about the number of substitutions of players during each match, then how many possible cases are there within the total time for every player on the field?

Let $x_1,x_2,...,x_k$ be a sequence of integers. A rearrangement of this sequence (the numbers in the sequence listed in some other order) is called a [b]scramble[/b] if no number in the new sequence is equal to the number originally in its location. For example, if the original sequence is $1,3,3,5$ then $3,5,1,3$ is a scramble, but $3,3,1,5$ is not. A rearrangement is called a [b]two-two[/b] if exactly two of the numbers in the new sequence are each exactly two more than the numbers that originally occupied those locations. For example, $3,5,1,3$ is a two-two of the sequence $1,3,3,5$ (the first two values $3$ and $5$ of the new sequence are exactly two more than their original values $1$ and $3$). Let $n\geq 2$. Prove that the number of scrambles of $1,1,2,3,...,n-1,n$ is equal to the number of two-twos of $1,2,3,...,n,n+1$. (Notice that both sequences have $n+1$ numbers, but the first one contains two 1s.)