The set of nonempty integers $A$ is said to be "elegant" if it is for any $a\in A,$ $1\leq k\leq 2023,$ $$\left| \left\{ b\in A:\left\lfloor\frac b{3^k}\right\rfloor =\left\lfloor\frac a{3^k}\right\rfloor\right\}\right| =2^k.$$ Prove that if the intersection of the integer set $S$ and any "elegant" set is not empty$,$ then $S$ contains an "elegant" set.

At a gala banquet, $12n + 6$ chairs, where $n \in N$, are equally arranged around a large round table. A seating will be called a proper seating of rank $n$ if a gathering of $6n + 3$ married couples sit around this table such that each seated person also has exactly one sibling (brother/sister) of the opposite gender present (siblings cannot be married to each other) and each man is seated closer to his wife than his sister. Among all proper seats of rank n find the maximum possible number of women seated closer to their brother than their husband. (The maximum is taken not only across all possible seating arrangements for a given gathering, but also across all possible gatherings.)

There are 2023 dice on the table. For 1 dollar, one can pick any dice and put it back on any of its four (other than top or bottom) side faces. How many dollars at a minimum will guarantee that all the dice have been repositioned to show equal number of dots on top faces? [i]Egor Bakaev[/i]

Alzim and Badril are playing a game on a hexagonal lattice grid with 37 points (4 points a side), all of them uncolored. On his turn, Alzim colors one uncolored point with the color red, and Badril colors [b]two[/b] uncolored points with the color blue. The game ends either when there is an equilateral triangle whose vertices are all red, or all points are colored. If the former happens, then Alzim wins, otherwise Badril wins. If Alzim starts the game, does Alzim have a strategy to guarantee victory?

Initially, the numbers $1, 2, \dots, 2024$ are written on a blackboard. Trixi and Nana play a game, taking alternate turns. Trixi plays first. The player whose turn it is chooses two numbers $a$ and $b$, erases both, and writes their (possibly negative) difference $a-b$ on the blackboard. This is repeated until only one number remains on the blackboard after $2023$ moves. Trixi wins if this number is divisible by $3$, otherwise Nana wins. Which of the two has a winning strategy? [i](Birgit Vera Schmidt)[/i]

Let $n$ and $k$ be integers such that $0\leq k\leq n$. Prove that $\sum_{u=0}^{k}\binom{n+u-1}{u}\binom{n}{k-2u}=\binom{n+k-1}{k}$. Note that we use the following conventions: $\binom{r}{0}=1$ for every integer $r$; $\binom{u}{v}=0$ if $u$ is a nonnegative integer and $v$ is an integer satisfying $v<0$ or $v>u$. Darij

There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.

There are $11$ committees in a club. Each committee has $5$ members and every two committees have a member in common. Show that there is a member who belongs to $4$ committees.

On the vertices of a convex $ n$-gon are put $ m$ stones, $ m > n$. In each move we can choose two stones standing at the same vertex and move them to the two distinct adjacent vertices. After $ N$ moves the number of stones at each vertex was the same as at the beginning. Prove that $ N$ is divisible by $ n$.

In a given group of people $\mathcal{F}$, each member has at least two acquaintances from $\mathcal{F}$. Moreover, for each cycle $A_{1} \leftrightarrow A_{2} \leftrightarrow ... \leftrightarrow A_{n} \leftrightarrow A_{1}$ in $\mathcal{F}$ (here '$X \leftrightarrow Y$' means that $X$ and $Y$ are acquaintances), each $A_i$ knows exactly two other members $A_j$ of the cycle. Prove that there exist $X, Y \in \mathcal{F}$ such that each of them has exactly two acquaintances in $\mathcal{F}$, and $X, Y$ have at least one common acquaintance in the group. [i]Authored by Mirko Petrusevski[/i]

There are $99$ space stations. Each pair of space stations is connected by a tunnel. There are $99$ two-way main tunnels, and all the other tunnels are strictly one-way tunnels. A group of $4$ space stations is called [i]connected[/i] if one can reach each station in the group from every other station in the group without using any tunnels other than the $6$ tunnels which connect them. Determine the maximum number of connected groups.

Freddy writes down numbers $1, 2,\ldots ,n$ in some order. Then he makes a list of all pairs $(i, j)$ such that $1\le i<j\le n$ and the $i$-th number is bigger than the $j$-th number in his permutation. After that, Freddy repeats the following action while possible: choose a pair $(i, j)$ from the current list, interchange the $i$-th and the $j$-th number in the current permutation, and delete $(i, j)$ from the list. Prove that Freddy can choose pairs in such an order that, after the process finishes, the numbers in the permutation are in ascending order.

Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Ana and Beto are playing a game. Ana writes a whole number on the board. Beto then has the right to erase the number and add $2$ to it, or erase the number and subtract $3$, as many times as he wants. Beto wins if he can get $2021$ after a finite number of stages; otherwise, Ana wins. Which player has a winning strategy?

Haydn picks two different integers between $1$ and $100$, inclusive, uniformly at random. The probability that their product is divisible by $4$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Several frogs are sitting on the real line at distinct integer points. In each move, one of them can take a $1$-jump towards the right as long as they are still in on distinct points. We calculate the number of ways they can make $N$ moves in this way for a positive integer $N$. Prove that if the jumps were all towards the left, we will still get the same number of ways. [i](F. Petrov)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

Let $n\geqslant3$ be an integer. Let $A_1A_2\ldots A_n$ be a convex $n$-gon. Consider a line $g$ through $A_1$ that does not contain a further vertice of the $n$-gon. Let $h$ be the perpendicular to $g$ through $A_1$. Project the $n$-gon orthogonally on $h$. For $j=1,\ldots,n$, let $B_j$ be the image of $A_j$ under this projection. The line $g$ is called admissible if the points $B_j$ are pairwise distinct. Consider all convex $n$-gons and all admissible lines $g$. How many different orders of the points $B_1,\ldots,B_n$ are possible?

The rows and columns of a $2^n \times 2^n$ table are numbered from $0$ to $2^{n}-1.$ The cells of the table have been coloured with the following property being satisfied: for each $0 \leq i,j \leq 2^n - 1,$ the $j$-th cell in the $i$-th row and the $(i+j)$-th cell in the $j$-th row have the same colour. (The indices of the cells in a row are considered modulo $2^n$.) Prove that the maximal possible number of colours is $2^n$. [i]Proposed by Hossein Dabirian, Sepehr Ghazi-nezami, Iran[/i]

Each of a pair of opposite faces of a unit cube is marked by a dot. Each of another pair of opposite faces is marked by two dots. Each of the remaining two faces is marked by three dots. Eight such cubes are used to construct a $2\times 2 \times 2$ cube. Count the total number of dots on each of its six faces. Can we obtain six consecutive numbers? (A Shapovalov)

Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$. Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$. Find a closed form for $a_n$. [i]Proposed by Bobby Shen[/i]

Find all positive integers $n$ such that the unit segments of an $n \times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.

Johnny once saw plums hanging, like eggs so big and numbered according to the first natural numbers. He is the first to pick the plum with number $2$. After that, Jantje picks the plum each time with the smallest number $n$ that satisfies the following two conditions: $\bullet$ $n$ is greater than all numbers on the already picked plums, $\bullet$ $n$ is not the product of two equal or different numbers on already picked plums. We call the numbers on the picked plums plum numbers. Is $100 000$ a plum number? Justify your answer.