In a plane, there is an infinite triangular grid consists of equilateral triangles whose lengths of the sides are equal to $ 1$, call the vertices of the triangles the lattice points, call two lattice points are adjacent if the distance between the two points is equal to $ 1;$ A jump game is played by two frogs $ A,B,$ "A jump" is called if the frogs jump from the point which it is lying on to its adjacent point, " A round jump of $ A,B$" is called if first $ A$ jumps and then $ B$ by the following rules: Rule (1): $ A$ jumps once arbitrarily, then $ B$ jumps once in the same direction, or twice in the opposite direction; Rule (2): when $ A,B$ sits on adjacent lattice points, they carry out Rule (1) finishing a round jump, or $ A$ jumps twice continually, keep adjacent with $ B$ every time, and $ B$ rests on previous position; If the original positions of $ A,B$ are adjacent lattice points, determine whether for $ A$ and $ B$,such that the one can exactly land on the original position of the other after a finite round jumps.

Determine the number of arrangements $ a_1,a_2,...,a_{10}$ of the numbers $ 1,2,...,10$ such that $ a_i>a_{2i}$ for $ 1 \le i \le 5$ and $ a_i>a_{2i\plus{}1}$ for $ 1 \le i \le 4$.

A set $X=\{ x_1,x_2,\ldots,x_n \}$ consisting of $n$ positive integers is given. It is known that the greatest common divisor of any four different elements of $X$ is $1$. For every number $x_i$ let $m_i$ be the number of elements of $X$, which are divisible by $x_i$ For every $n \geq 4$, find the maximal possible value of the sum $m_1+\ldots+m_n$ [i]A. Vaidzelevich[/i]

On circle in clockwise order are written positive integers from $1$ to $2010$. Let us cross out number $1$, then number $10$, then number $19$, and so on every $9$th number in that direction. Which number will be first crossed out twice? How many numbers at that moment are not crossed out?

A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities there is a route (a sequence of flights) using only one of the airplane companies?

A $4 \times 4$ board has all its squares painted white. An allowed operation is to choose a rectangle that contains $3$ squares and paint each of the boxes as follows: a) If the box is white then it is painted black. b) If the box is black then it is painted white. Prove that by applying the allowed operation several times, it is impossible get the entire board painted black.

Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$

There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?

In there $2018\times 2018$ square cells colored in white or black. It is known, that exists $10 \times 10$ square with only white cells and $10\times 10$ square with only black cells. For what minimal $d$ always exists square $10\times 10$ such that the number of black and white cells differs by no more than $d$?

There are $12$ chairs arranged in a circle, numbered from $ 1$ to $ 12$. How many ways are there to select some of the chairs in such a way that our selection includes $3$ consecutive chairs somewhere?

The diagonals of a convex $18$-gon are colored in $5$ different colors, each color appearing on an equal number of diagonals. The diagonals of one color are numbered $1, 2,\cdots$. One randomly chooses one-fifth of all the diagonals. Find the number of possibilities for which among the chosen diagonals there exist exactly $n$ pairs of diagonals of the same color and with fixed indices $i, j$.

Daniel has an unlimited supply of tiles labeled “$2$” and “$n$” where $n$ is an integer. Find (with proof) all the values of $n$ that allow Daniel to fill an $8 \times 10$ grid with these tiles such that the sum of the values of the tiles in each row or column is divisible by $11$.

How many ways are there to fill a $3 \times 3$ grid with the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, and $9$, such that the set of three elements in every row and every column form an arithmetic progression in some order? (Each number must be used exactly once)

Prove that the set $\{1, 2, . . . , 1986\}$ can be partitioned into $27$ disjoint sets so that no one of these sets contains an arithmetic triple (i.e., three distinct numbers in an arithmetic progression).

Let $A_1A_2\cdots A_{2002}$ be a regular 2002 sided polygon. Each vertex $A_i$ is associated with a positive integer $a_i$ such that the following condition is satisfied: If $j_1,j_2,\cdots, j_k$ are positive integers such that $k<500$ and $A_{j_1}, A_{j_2}, \cdots A_{j_k}$ is a regular $k$ sided polygon, then the values of $a_{j_1},A_{j_2}, \cdots A_{j_k}$ are all different. Find the smallest possible value of $a_1+a_2+\cdots a_{2002}$

a) A game master divides a group of $40$ players into four teams of ten. The players do not know what the teams are, however the master gives each player a card containing the names of two other players: one of them is a teammate and the other is not, but the master does not tell the player which is which. Can the master write the names on the cards in such a way that the players can determine the teams? (All of the players can work together to do so.) b) On the next occasion, the game master writes the names of $7$ teammates and $2$ opposing players on each card (possibly in a mixed up order). Now he wants to write the names in such a way that the players together cannot determine the four teams. Is it possible for him to achieve this? c) Can he write the names in such a way that the players together cannot determine the four teams, if now each card contains the names of $6$ teammates and $2$ opposing players (possibly in a mixed up order)?

Let $n$ be a natural number. There are $n$ boys and $n$ girls standing in a line, in any arbitrary order. A student $X$ will be eligible for receiving $m$ candies, if we can choose two students of opposite sex with $X$ standing on either side of $X$ in $m$ ways. Show that the total number of candies does not exceed $\frac 13n(n^2-1).$

Find all positive integers $m,n$ such that the $m \times n$ grid can be tiled with figures formed by deleting one of the corners of a $2 \times 3$ grid. [i]usjl, ST[/i]

In a summer school, there are $n>4$ students. It is known that, among these students, i. If two ones are friends, then they don't have any common friends. ii If two ones are not friends, then they have exactly two common friends. 1. Prove that $8n-7$ must be a perfect square. 2. Determine the smallest possible value of $n$.

Around a round table the people $P_1, P_2,..., P_{2013}$ are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none); there are a total of $10000$ coins. Starting with $P_1$ and proceeding in clockwise order, each person does the following on their turn: [list][*]If they have an even number of coins, they give all of their coins to their neighbor to the left. [*]If they have an odd number of coins, they give their neighbor to the left an odd number of coins (at least $1$ and at most all of their coins) and keep the rest.[/list] Prove that, repeating this procedure, there will necessarily be a point where one person has all of the coins.

Given three colors and a rectangle m × n dice unit, we want to color each segment constituting one side of a square drive with one of three colors so that each square unit have two sides of one color and two sides another color. How many colorings we have?

Consider the set $M=\{1,2,3,...,2003\}$. How many subsets of $M$ with even number of elements exist?

$N$ men are not acquainted each other. You need to introduce some of them to some of them in such a way, that no three men will have equal number of acquaintances. Prove that it is possible for all $N$.

Let $A$ be a finite set, and $A_1,A_2,\cdots, A_n$ are subsets of $A$ with the following conditions: (1) $|A_1|=|A_2|=\cdots=|A_n|=k$, and $k>\frac{|A|}{2}$; (2) for any $a,b\in A$, there exist $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$ such that $a,b\in A_r\cap A_s\cap A_t$; (3) for any integer $i,j\, (1\leq i<j\leq n)$, $|A_i\cap A_j|\leq 3$. Find all possible value(s) of $n$ when $k$ attains maximum among all possible systems $(A_1,A_2,\cdots, A_n,A)$.

$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.