Let $a, b, c, p, q, r$ be positive integers with $p, q, r \ge 2$. Denote \[Q=\{(x, y, z)\in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b , 0 \le z \le c \}. \] Initially, some pieces are put on the each point in $Q$, with a total of $M$ pieces. Then, one can perform the following three types of operations repeatedly: (1) Remove $p$ pieces on $(x, y, z)$ and place a piece on $(x-1, y, z)$ ; (2) Remove $q$ pieces on $(x, y, z)$ and place a piece on $(x, y-1, z)$ ; (3) Remove $r$ pieces on $(x, y, z)$ and place a piece on $(x, y, z-1)$. Find the smallest positive integer $M$ such that one can always perform a sequence of operations, making a piece placed on $(0,0,0)$, no matter how the pieces are distributed initially.

We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots$,5 is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

$p>2$ persons participate at a chess tournament, two players play at most one game against each other. After $n$ games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other. Show that $n \leq \frac{p^{2}}4$.

A finite set $\mathcal{S}$ of positive integers is called cardinal if $\mathcal{S}$ contains the integer $|\mathcal{S}|$ where $|\mathcal{S}|$ denotes the number of distinct elements in $\mathcal{S}$. Let $f$ be a function from the set of positive integers to itself such that for any cardinal set $\mathcal{S}$, the set $f(\mathcal{S})$ is also cardinal. Here $f(\mathcal{S})$ denotes the set of all integers that can be expressed as $f(a)$ where $a \in \mathcal{S}$. Find all possible values of $f(2024)$ $\quad$ Proposed by Sutanay Bhattacharya

A circle is divided into $2n$ equal sectors, $n \in \mathbb{N}$. Vitya and Masha are playing the following game. At first, Vitya writes one number in every sector from the set $\{1,2,\ldots,n\}$ and every number is used exatly twice. After that Masha chooses $n$ consecutive sectors and writes $1$ in the first sector, $2$ in the second, $n$ in the last. Vitya wins if at least in one sector two equal number will be written, otherwise Masha wins. Find all $n$ for which Vitya can guarantee his win.

In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word. A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word. For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$. Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.

Six ducklings swim on the surface of a pond, which is in the shape of a circle with radius $5$ m. Show that at every moment, two of the ducklings swim on the distance of at most $5$ m from each other.

An ordinary domino tile can be identified as a pair $(k,m)$ where numbers $k$ and $m$ can get values $0, 1, 2, 3, 4, 5$ and $6.$ Pairs $(k,m)$ and $(m, k)$ determine the same tile. In particular, the pair $(k, k)$ determines one tile. We say that two domino tiles [i]match[/i], if they have a common component. [i]Generalized n-domino tiles[/i] $m$ and $k$ can get values $0, 1,... , n.$ What is the probability that two randomly chosen $n$-domino tiles match?

A convex$N-$gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new $2N$ segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones. [i]($8$ points)[/i]

Let $m > n$ be positive integers. In the host country of the International Olympiad in Informatics this year, there are coins of $1$, $2$, $\ldots$, $n$ [i]alexandrias[/i], [i]lira[/i] banknotes, each worth $m$ alexandrias, and [i]pharaoh[/i] banknotes, each worth $m+n$ alexandrias. Let $A$ be a positive integer. Boris wants to exchange the amount $A$ using coins and lira, using no more than $m-1$ coins, while Vesko wants to exchange the amount $A$ using coins and pharaohs, using no more than $m$ coins. Prove that regardless of the value of $A$, the number of ways for each of them to fulfill their desire is the same.

Let $300$ students participate to the Olympiad. Between each $3$ participants there is a pair that are not friends. Hamza enumerates participants in some order and denotes by $x_i$ the number of friends of $i$-th participant. It occurs that $\{x_1,x_2,...,x_{299},x_{300}\} = \{1, 2,..., N - 1,N\}$ Find the biggest possible value for $N$.

On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

The country of HMMTLand has $8$ cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly $3$ other cities via a single road, and from any pair of distinct cities, either exactly $0$ or $2$ other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.

We say that a set $A \subset \mathbb Z$ is irregular if, for any different elements $x, y \in A$, there is no element of the form $x + k(y -x)$ different from $x$ and $y$ (where $k$ is an integer). Is there an infinite irregular set?

At a party, $25$ elves give each other presents. No elf gives a present to herself. Each elf gives a present to at least one other elf, but no elf gives a present to all other elves. Show that it is possible to choose a group of three elves including at least two elves who give a present to exactly one of the other two elves in the group.

There are $18$ students in a class. Each student is asked two questions: how many other students have the same first name as you and how many other students have the same surname as you. The answers $0, 1, 2, . . ., 7$ all occur. Prove that there are two students with the same first name and last name.

$10$ teams participated in a football tournament: every two teams played each other exactly once. After the end of the tournament it turned out that all teams got different amount of points and some teams won more games, than the winner of the tournament, call them strong. What is the maximum number of teams that could be strong? (In football the winner of the match gets three points, the loser - 0 points, and if the match ends in a draw both teams get 1 point.)

Each positive integer should be colored red or green in such a way that the following two conditions are met: - Let $n$ be any red number. The sum of any $n$ (not necessarily different) red numbers is red. - Let $m$ be any green number. The sum of any $m$ (not necessarily different) green numbers is green. Determine all such colorings.

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal S$ are collinear. A [i]windmill[/i] is a process that starts with a line $\ell$ going through a single point $P \in \mathcal S$. The line rotates clockwise about the [i]pivot[/i] $P$ until the first time that the line meets some other point belonging to $\mathcal S$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal S$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal S$ as a pivot infinitely many times. [i]Proposed by Geoffrey Smith, United Kingdom[/i]

In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off). Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.

On a table, there are $16$ weights of the same appearance, which have all the integer weights from $13$ to $28$ grams, that is, they weigh $13, 14, 15, \dots, 28$ grams. Determine the four weights that weigh $13, 14, 27, 28$ grams, using a two-pan balance at most $26$ times.

Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any $x$-meter portion of the wire, there is an $\tfrac{x}{10}$ probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest neighbor?

Alice and Bob play on an infinite board formed by equilateral triangles. In each turn, Alice first places a white token on an unoccupied cell, and then Bob places a black token on an unoccupied cell. Alice's goal is to eventually have $k$ white tokens on a line. Determine the maximum value of $k$ for which Alice can achieve this no matter how Bob plays. [i]Proposed by Oriol Solé[/i]

The numbers $1,\frac12,\frac13,\ldots,\frac1{1999}$ are written on a blackboard. In every step, we choose two of them, say $a$ and $b$, erase them, and write the number $ab+a+b$ instead. This step is repeated until only one number remains. Can the last remaining number be equal to $2000$?

Let $d_1, d_2, \ldots, d_n$ be given integers. Show that there exists a graph whose sequence of degrees is $d_1, d_2, \ldots, d_n$ and which contains an perfect matching if, and only if, there exists a graph whose sequence of degrees is $d_2, d_2, \ldots, d_n$ and a graph whose sequence of degrees is $d_1-1, d_2-1, \ldots, d_n-1$.