A grid consists of all points of the form $(m, n)$ where $m$ and $n$ are integers with $|m|\le 2019,|n| \le 2019$ and $|m| +|n| < 4038$. We call the points $(m,n)$ of the grid with either $|m| = 2019$ or $|n| = 2019$ the [i]boundary points[/i]. The four lines $x = \pm 2019$ and $y= \pm 2019$ are called [i]boundary lines[/i]. Two points in the grid are called [i]neighbours [/i] if the distance between them is equal to $1$. Anna and Bob play a game on this grid. Anna starts with a token at the point $(0,0)$. They take turns, with Bob playing first. 1) On each of his turns. Bob [i]deletes [/i] at most two boundary points on each boundary line. 2) On each of her turns. Anna makes exactly three [i]steps[/i] , where a [i]step [/i] consists of moving her token from its current point to any neighbouring point, which has not been deleted. As soon as Anna places her token on some boundary point which has not been deleted, the game is over and Anna wins. Does Anna have a winning strategy? [i]Proposed by Demetres Christofides, Cyprus[/i]

On a table, cards numbered $1, 2, \ldots , 200$ are laid out in a row in some order, and a line is drawn on the table between some two of them. It is allowed to swap two adjacent cards if the number on the left is greater than the number on the right. After a few such moves, the cards were arranged in ascending order. Prove we have swapped pairs of cards separated by the line no more than 1884 times.

The closed broken line $M$ has odd number of vertices -- $A_1,A_2,..., A_{2n+1}$ in sequence. Let us denote with $S(M)$ a new closed broken line with vertices $B_1,B_2,...,B_{2n+1}$ -- the midpoints of the first line links: $B_1$ is the midpoint of $[A_1A_2], ... , B_{2n+1}$ -- of $[A_{2n+1}A_1]$. Prove that in a sequence $M_1=S(M), ... , M_k = S(M_{k-1}), ...$ there is a broken line, homothetic to the $M$.

In a school, every pair of students are either friends or strangers. Friendship is mutual, and no student is friends with themselves. A sequence of (not necessarily distinct) students $A_1, A_2, \dots, A_{2023}$ is called [i]mischievous[/i] if $\bullet$ Total number of friends of $A_1$ is odd. $\bullet$ $A_i$ and $A_{i+1}$ are friends for $i=1, 2, \dots, 2022$. $\bullet$ Total number of friends of $A_{2023}$ is even. Prove that the total number of [i]mischievous[/i] sequences is even.

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

The squares of a $10 \times 10$ chessboard are labelled with $1,2,...,100 $ in the usual way: the $i$-th row contains the numbers $10i -9,10i - 8,...,10i$ in increasing order. The signs of fifty numbers are changed so that each row and each column contains exactly five negative numbers. Show that after this change the sum of all numbers on the chessboard is zero.

Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive.

Let $S$ be a set of $n^2 + 1$ closed intervals ($n$ a positive integer). Prove that at least one of the following assertions holds: [b](i)[/b] There exists a subset $S'$ of $n+1$ intervals from $S$ such that the intersection of the intervals in $S'$ is nonempty. [b](ii)[/b] There exists a subset $S''$ of $n + 1$ intervals from $S$ such that any two of the intervals in $S''$ are disjoint.

Let $n \geq 2$ be a positive integer. At the center of a circular garden is a guard tower. On the outskirt of the garden there are $n$ garden dwarfs regularly spaced. In the tower are attentive supervisors. Each supervisor controls a portion of the garden delimited by two dwarfs. We say that the supervisor $A$ controls the supervisor $B$ if the region of $B$ is contained in that of $A$. Among the supervisors there are two groups: the apprentices and the teachers. Each apprentice is controlled by exactly one teachers, and controls no one, while the teachers are not controlled by anyone. The entire garden has the following maintenance costs: - One apprentice costs 1 gold per year. - One teacher costs 2 gold per year. - A garden dwarf costs 2 gold per year. Show that the garden dwarfs cost at least as much as the supervisors.

A society has to elect a board of governors. Each member of the society has chosen $10$ candidates for the board, but he will be happy if at least one of them will be on the board. For each six members of the society there exists a board consisting of two persons making all of these six members happy. Prove that a board consisting of $10$ persons can be elected making every member of the society happy.

There are 2 pizzerias in a town, with 2010 pizzas each. Two scientists $A$ and $B$ are taking turns ($A$ is first), where on each turn one can eat as many pizzas as he likes from one of the pizzerias or exactly one pizza from each of the two. The one that has eaten the last pizza is the winner. Which one of them is the winner, provided that they both use the best possible strategy?

How many ways are there to express $2548$ as a sum of at least two positive integers, where two sums that differ in order are considered different?

Let $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$. Determine $\displaystyle n$ sets $\displaystyle A_i$, $\displaystyle 1 \leq i \leq n$, from the plane, pairwise disjoint, such that: (a) for every circle $\displaystyle \mathcal C$ from the plane and for every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$ we have $\displaystyle A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi$; (b) for all lines $\displaystyle d$ from the plane and every $\displaystyle i \in \left\{ 1,2,\ldots,n \right\}$, the projection of $\displaystyle A_i$ on $\displaystyle d$ is not $\displaystyle d$.

There are $n$ clubs composed of $4$ students out of all $9$ students. For two arbitrary clubs, there are no more than $2$ students who are a member of both clubs. Prove that $n\le 18$. Translator’s Note. We can prove $n\le 12$, and we can prove that the bound is tight. (Credits to rkm0959 for translation and document)

Let $n \geq 2$ be a given integer. At any point $(i, j)$ with $i, j \in\mathbb{ Z}$ we write the remainder of $i+j$ modulo $n$. Find all pairs $(a, b)$ of positive integers such that the rectangle with vertices $(0, 0)$, $(a, 0)$, $(a, b)$, $(0, b)$ has the following properties: [b](i)[/b] the remainders $0, 1, \ldots , n-1$ written at its interior points appear the same number of times; [b](ii)[/b] the remainders $0, 1, \ldots , n -1$ written at its boundary points appear the same number of times.

Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minimum integer $x$ with proof such that if $n-m\geq x,$ then Pratyya's number will be larger than Payel's number everyday.

A polyomino is a figure which consists of unit squares joined together by their sides. (A polyomino may contain holes.) Let $n\ge3$ be a positive integer. Consider a grid of unit square cells which extends to infinity in all directions. Find, in terms of $n$, the greatest positive integer $C$ which satisfies the following condition: For every colouring of the cells of the grid in $n$ colours, there is some polyomino within the grid which contains at most $n-1$ colours and whose area is at least $C$. Proposed by Nikolai Beluhov, Stara Zagora, Bulgaria and Stefan Gerdjikov, Sofia, Bulgaria

In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied? [i]Proposed by Gerhard Wöginger, Austria[/i]

One hundred sages play the following game. They are waiting in some fixed order in front of a room. The sages enter the room one after another. When a sage enters the room, the following happens - the guard in the room chooses two arbitrary distinct numbers from the set {$1,2,3$}, and announces them to the sage in the room. Then the sage chooses one of those numbers, tells it to the guard, and leaves the room, and the next enters, and so on. During the game, before a sage chooses a number, he can ask the guard what were the chosen numbers of the previous two sages. During the game, the sages cannot talk to each other. At the end, when everyone has finished, the game is considered as a failure if the sum of the 100 chosen numbers is exactly $200$; else it is successful. Prove that the sages can create a strategy, by which they can win the game.

Let \( n, k \geq 1 \). In a school, there are \( n \) students and \( k \) clubs. Each student participates in at least one of the clubs. One day, a school uniform was established, which could be either blue or red. Each student chose only one of these colors. Every day, the principal visited one of the clubs, forcing all the students in it to switch the colors of the uniforms they wore. Assuming that the students are distributed in clubs in such a way that any initial choice of uniforms they make, after a certain number of days, it is possible to have at most one student with one of the colors. Show that \[ n \geq 2^{n-k-1} - 1. \]

Find maximal value of positive integer $n$ such that there exists subset of $S=\{1,2,...,2001\}$ with $n$ elements, such that equation $y=2x$ does not have solutions in set $S \times S$

A graph has $n$ points and $\frac{n(n-1)}{2}$ edges. Each edge is colored with one of $k$ colors so that there are no closed monochrome paths. What is the largest possible value of $n$ (given $k$)?

In a school with $101$ students, each student has at least one friend among the other students. Show that for every integer $1<n<101$, a group of $n$ students can be selected from this school in such a way that each selected student has at least one friend among the other selected students.

Let m be a positive integer, and let $S=\{a_1=1, a_2, ..., a_m\}$ be a set of positive integers. Prove that there exists a positive integer $n$ with $n\le m$ and a set $T={b_1, b_2, ..., b_n}$ of positive integers such that (a) all the subsets of $T$ have distinct sums of elements; (b) each of the numbers $a_1$, $a_2$, ..., $a_m$ is the sum of the elements of a subset of $T$.

Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red. Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$ [I]Netherlands[/i]