An $ n \times n, n \geq 2$ chessboard is numbered by the numbers $ 1, 2, \ldots, n^2$ (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least $ n.$

Adel draws an $m \times n$ grid of dots on the coordinate plane, at the points of integer coordinates $(a,b)$ where $1 \le a \le m$ and $1 \le b \le n$. He proceeds to draw a closed path along $k$ of these dots, $(a_1, b_1)$,$(a_2,b_2)$,...,$(a_k,b_k)$, such that $(a_i,b_i)$ and $(a_{i+1}, b_{i+1})$ (where $(a_{k+1}, b_{k+1}) = (a_1, b_1)$) are $1$ unit apart for each $1 \le i \le k$. Adel makes sure his path does not cross itself, that is, the $k$ dots are distinct. Find, with proof, the maximum possible value of $k$ in terms of $m$ and $n$.

Among 6 coins one is counterfeit (its weight differs from that real one and neither weights is known). Using scales that show the total weight of coins placed on the cup, find the counterfeit coin in 3 weighings. [i](5 points)[/i]

Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$. Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$

Frog is in the origin of decartes coordinate system. Every second frog jumpes horizontally or vertically in some of the $4$ adjacent points which coordinates are integers. Find number of different points in which frog can be found in $2015$ seconds.

$25$ checkers are placed on $25$ leftmost squares of $1 \times N$ board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal $N$ such that all the checkers could be placed in the row of $25$ successive squares but in the reverse order.

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

Consider all words constituted by eight letters from $\{C ,H,M, O\}$. We arrange the words in an alphabet sequence. Precisely, the first word is $CCCCCCCC$, the second one is $CCCCCCCH$, the third is $CCCCCCCM$, the fourth one is $CCCCCCCO, ...,$ and the last word is $OOOOOOOO$. a) Determine the $2017$th word of the sequence? b) What is the position of the word $HOMCHOMC$ in the sequence?

By one magic nut, Wicked Witch can either turn a flea into a beetle or a spider into a bug; while by one magic acorn, she can either turn a flea into a spider or a beetle into a bug. In the evening Wicked Witch had spent 20 magic nuts and 23 magic acorns. By these actions, the number of beetles increased by 5. Determine what was the change in the number of spiders. (Find all possible answers and prove that the other answers are impossible.)

A plane is cut into unit squares, which are then colored in $n$ colors. A polygon $P$ is created from $n$ unit squares that are connected by their sides. It is known that any cell polygon created by $P$ with translation, covers $n$ unit squares in different colors. Prove that the plane can be covered with copies of $P$ so that each cell is covered exactly once.

Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

There is a rectangular sheet of paper on an infinite blackboard. Marvin secretly chooses a convex $2024$-gon $P$ that lies fully on the piece of paper. Tigerin wants to find the vertices of $P$. In each step, Tigerin can draw a line $g$ on the blackboard that is fully outside the piece of paper, then Marvin replies with the line $h$ parallel to $g$ that is the closest to $g$ which passes through at least one vertex of $P$. Prove that there exists a positive integer $n$, independent of the choice of the polygon, such that Tigerin can always determine the vertices of $P$ in at most $n$ steps.

We are given $30$ boots standing in a row, $15$ of which are for right feet and $15$ for the left. Prove that there are ten successive boots somewhere in this row with $5$ right and $5$ left boots among them. (D. Fomin, Leningrad)

Dedalo buys a finite number of binary strings, each of finite length and made up of the binary digits 0 and 1. For each string, he pays $(\frac{1}{2})^L$ drachmas, where $L$ is the length of the string. The Minotaur is able to escape the labyrinth if he can find an infinite sequence of binary digits that does not contain any of the strings Dedalo bought. Dedalo’s aim is to trap the Minotaur. For instance, if Dedalo buys the strings $00$ and $11$ for a total of half a drachma, the Minotaur is able to escape using the infinite string $01010101 \ldots$. On the other hand, Dedalo can trap the Minotaur by spending $75$ cents of a drachma: he could for example buy the strings $0$ and $11$, or the strings $00, 11, 01$. Determine all positive integers $c$ such that Dedalo can trap the Minotaur with an expense of at most $c$ cents of a drachma.

Find the sum of the answers to all even numbered Short Answer problems, with the exception of #26, rounded to the nearest tenth. [i](.7 points)[/i]

A cube has one of its vertices and all edges connected to that vertex deleted. How many ways can the letters from the word "$AMONGUS$" be placed on the remaining vertices of the cube so that one can walk along the edges to spell out "$AMONGUS$"? Note that each vertex will have at most $1$ letter, and one vertex is deleted and not included in the walk

A rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles, and similar to the large rectangle, so that the sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures $1$. Determine what is the minimum possible value of the area of the large rectangle.

$2019$ coins are on the table. Two students play the following game making alternating moves. The first player can in one move take the odd number of coins from $ 1$ to $99$, the second player in one move can take an even number of coins from $2$ to $100$. The player who can not make a move is lost. Who has the winning strategy in this game?

Pedro and Tiago are playing a game with a deck of n cards, numbered from $1$ to $n$. Starting with Pedro, they choose cards alternately, and receive the number of points indicated by the cards. However, whenever the player chooses the card with the highest number among those remaining in the deck, he is forced to pass his next turn, not choosing any card. When the deck runs out, the player with the most points wins. Knowing that Tiago can at least draw, regardless of Pedro's moves, how many cards are in the deck? Indicates all possibilities,

Let a board game has $10$ cards: $3$ [b]skull[/b] cards, $5$ [b]coin[/b] cards and $2$ [b]blank[/b] cards. We put these $10$ cards downward and shuffle them and take cards one by one from the top. Once $3$ [b]skull[/b] cards or [b]coin[/b] cards appears we stop. What is the possibility of it stops because there appears $3$ [b]skull[/b] cards?

A $1$ or $0$ is placed on each square of a $4 \times 4$ board. One is allowed to change each symbol in a row, or change each symbol in a column, or change each symbol in a diagonal (there are $14$ diagonals of lengths $1$ to $4$). For which arrangements can one make changes which end up with all $0$s?

Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.

There are exactly 7 possible tetrominos (groups of 4 connected squares in a grid): [img]https://cdn.discordapp.com/attachments/813077401265242143/816189385859006474/tetris.png[/img] Daniel has a $2 \times 20210$ rectangle and wants to tile the interior with tetrominos without overlaps, pieces sticking out, or extra pieces left over. Note that you are allowed to rotate tetrominos but not reflect them. For how many multisets of tetrominos (ie. an ordered tuple of how many of each tile he has) is it possible to exactly tile his $2\times20210$ rectangle? [i]Proposed by Dilhan Salgado[/i]

Let $n \geq 6$ and $3 \leq p < n - p$ be two integers. The vertices of a regular $n$-gon are colored so that $p$ vertices are red and the others are black. Prove that there exist two congruent polygons with at least $[p/2] + 1$ vertices, one with all the vertices red and the other with all the vertices black.

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]