There are $9$ vertical columns in a row. In some places, horizontal sticks are inserted between adjacent columns, no two are at the same height. The beetle crawls from the bottom up; when he meets the wand, he crawls over it to the next column and continues to crawl up. It is known that if a beetle starts at the bottom of the first (leftmost) column, then it will end its journey on the ninth (rightmost) column. Is it always possible to remove one stick so that the beetle ends up at the top of the fifth column? (For example, if the sticks are arranged as in picture, the beetle will crawl along a solid line. If you remove the third one A stick in the path of the beetle, then it will crawl along the dotted line.) [i] Proposed by G. Karavaev[/i]

Consider a $2n\times 2n$ chessboard with all the $4n^2$ cells being white to start with. The following operation is allowed to be performed any number of times: "Three consecutive cells (in a row or column) are recolored (a white cell is colored black and a black cell is colored white)." Find all possible values of $n\ge 2$ for which using the above operation one can obtain the normal chess coloring of the given board.

A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.

Let $ABCD$ be a rectangle consisting of unit squares. All vertices of these unit squares inside the rectangle and on its sides have been colored in four colors. Additionally, it is known that: [list] [*] every vertex that lies on the side $AB$ has been colored in either the $1.$ or $2.$ color; [*] every vertex that lies on the side $BC$ has been colored in either the $2.$ or $3.$ color; [*] every vertex that lies on the side $CD$ has been colored in either the $3.$ or $4.$ color; [*] every vertex that lies on the side $DA$ has been colored in either the $4.$ or $1.$ color; [*] no two neighboring vertices have been colored in $1.$ and $3.$ color; [*] no two neighboring vertices have been colored in $2.$ and $4.$ color. [/list] Notice that the constraints imply that vertex $A$ has been colored in $1.$ color etc. Prove that there exists a unit square that has all vertices in different colors (in other words it has one vertex of each color).

Ten million fireflies are glowing in $\mathbb{R}^3$ at midnight. Some of the fireflies are friends, and friendship is always mutual. Every second, one firefly moves to a new position so that its distance from each one of its friends is the same as it was before moving. This is the only way that the fireflies ever change their positions. No two fireflies may ever occupy the same point. Initially, no two fireflies, friends or not, are more than a meter away. Following some finite number of seconds, all fireflies find themselves at least ten million meters away from their original positions. Given this information, find the greatest possible number of friendships between the fireflies. [i]Nikolai Beluhov[/i]

Let $P$ be a regular $n$-gon $A_1A_2\ldots A_n$. Find all positive integers $n$ such that for each permutation $\sigma (1),\sigma (2),\ldots ,\sigma (n)$ there exists $1\le i,j,k\le n$ such that the triangles $A_{i}A_{j}A_{k}$ and $A_{\sigma (i)}A_{\sigma (j)}A_{\sigma (k)}$ are both acute, both right or both obtuse.

$ 2^n $ coins are given to a couple of kids. Interchange of the coins occurs when some of the kids has at least half of all the coins. Then from the coins of one of those kids to the all other kids are given that much coins as the kid already had. In case when all the coins are at one kid there is no possibility for interchange. What is the greatest possible number of consecutive interchanges? ($ n $ is natural number)

Consider the set $X =\{1, 2,3, ...,2018\}$. How many positive integers $k$ with $2 \le k \le 2017$ that satisfy the following conditions: i) There exists some partition of the set $X$ into $1009$ disjoint pairs which are $(a_1, b_1),(a_2, b_2), ...,(a_{1009}, b_{1009})$ with $|a_i - b_i| \in \{1, k\}$. ii) For all partitions satisfy the condition (i), the sum $T = \sum^{1009}_{i=1} |a_i - b_i|$ has the right most digit is $9$

Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.

Let $\pi(n)$ denote the number of primes less than or equal to $n$. A subset of $S=\{1,2,\ldots, n\}$ is called [i]primitive[/i] if there are no two elements in it with one of them dividing the other. Prove that for $n\geq 5$ and $1\leq k\leq \pi(n)/2,$ the number of primitive subsets of $S$ with $k+1$ elements is greater or equal to the number of primitive subsets of $S$ with $k$ elements. [i]Proposed by Cs. Sándor, Budapest[/i]

Of $4n$ points in a row, $2n$ are colored white and $2n$ are colored black. Swot tha tthere are $2n$ consecutive points of which exactly $n$ are white and $n$ are black.

There are $200$ boxes on the table. In the beginning, each of the boxes contains a positive integer (the integers are not necessarily distinct). Every minute, Alice makes one move. A move consists of the following. First, she picks a box $X$ which contains a number $c$ such that $c = a + b$ for some numbers $a$ and $b$ which are contained in some other boxes. Then she picks a positive integer $k > 1$. Finally, she removes $c$ from $X$ and replaces it with $kc$. If she cannot make any mobes, she stops. Prove that no matter how Alice makes her moves, she won't be able to make infinitely many moves.

An $11\times 11$ chess board is covered with one $\boxed{ }$ shaped and forty $\boxed{ }\boxed{ }\boxed{ }$ shaped tiles. Determine the squares where $\boxed{}$ shaped tile can be placed.

The numbers $3$, $4$ ,$...$, $2019$ are written on a blackboard. David and Edgardo play alternately, starting with David. On their turn, each player must erase a number from the board and write two positive integers whose sum is equal to the number just deleted. The winner is the one who makes all the numbers on the board equal. Determine who has a winning strategy and describe it.

Determine how many $100$-positive integer sequences satisfy the two conditions following: - At least one term of the sequence is equal to $4$ or $5$. - Any two adjacent terms differ as a maximum in $2$.

Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that (a) $a_1 + a_2 + .. + a_n \ge n^2$ and (b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$

Let $x_0 = x_{101} = 0$. The numbers $x_1, x_2,...,x_{100}$ are chosen at random from the interval $[0, 1]$ uniformly and independently. Compute the probability that $2x_i \ge x_{i-1} + x_{i+1}$ for all $i = 1, 2,..., 100.$

For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically? Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.

We have a pile of $2016$ cards and a hat. We take out one card, put it in the hat and then divide the remaining cards into two arbitrary non empty piles. In the next step, we choose one of the two piles, we move one card from this pile to the hat and then divide this pile into two arbitrary non empty piles. This procedure is repeated several times : in the $k$-th step $(k>1)$ we move one card from one of the piles existing after the step $(k-1)$ to the hat and then divide this pile into two non empty piles. Is it possible that after some number of steps we get all piles containing three cards each?

(a) Two identical cogwheels with $14$ teeth each are given . One is laid horizontally on top of the other in such a way that their teeth coincide (thus the projections of the teeth on the horizontal plane are identical ) . Four pairs of coinciding teeth are cut off. Is it always possible to rotate the two cogwheels with respect to each other so that their common projection looks like that of an entire cogwheel? (The cogwheels may be rotated about their common axis, but not turned over.) (b) Answer the same question , but with two $13$-tooth cogwheels and four pairs of cut-off teeth.

The figure shows a $2\times 2$ grid that has been filled with the numbers $a, b, c$, and $d$. We say that this grid is [i]ordered[/i] if it is true that $a > b > c > d$ or that $a > d > c > b$. $\begin{tabular}{|l|l|} \hline a & b \\ \hline d & c \\ \hline \end{tabular}$ In how many ways can the numbers from $1$ to $1000$ be arranged in the cells of a $2 \times 500$ grid ($2$ rows and $500$ columns) so that each $2 \times 2$ sub-grid is ordered?

Let $ 1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $ \sum_{i\equal{}1}^n 1/a_i \leq 1$. Let $ y$ denote the number of positive integers smaller that $ x$ not divisible by any of the $ a_i$. Prove that \[ y > \frac{cx}{\log x}\] with a suitable positive constant $ c$ (independent of $ x$ and the numbers $ a_i$). [i]I. Z. Ruzsa[/i]

Three red books, three white books, and three blue books are randomly stacked to form three piles of three books each. The probability that no book is the same color as the book immediately on top of it is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

We call an integer $ n > 1$ [i]good[/i] if, for any natural numbers $ 1 \le b_1, b_2, \ldots , b_{n\minus{}1} \le n \minus{} 1$ and any $ i \in \{0, 1, \ldots , n \minus{} 1\}$, there is a subset $ I$ of $ \{1, \ldots , n \minus{} 1\}$ such that $ \sum_{k\in I} b_k \equiv i \pmod n$. (The sum over the empty set is zero.) Find all good numbers.

A group of $6$ students decided to make study groups and service activity groups according to the following principle: Each group must have exactly $3$ members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members. Supposing there are at least one group and no three students belong to the same study group and service activity group, prove that the minimum number of groups is $8$.