Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?

Let $n$ be a positive integer. The Georgian folk dance team consists of $2n$ dancers, with $n$ males and $n$ females. Each dancer, both male and female, is assigned a number from 1 to $n$. During one of their dances, all the dancers line up in a single line. Their wish is that, for every integer $k$ from 1 to $n$, there are exactly $k$ dancers positioned between the $k$th numbered male and the $k$th numbered female. Prove the following statements: a) If $n \equiv 1 \text{ or } 2 \mod{4}$, then the dancers cannot fulfill their wish. b) If $n \equiv 0 \text{ or } 3 \mod{4}$, then the dancers can fulfill their wish. [i]Proposed by Giorgi Arabidze, Georgia[/i]

We are given a certain number of identical sets of weights; each set consists of four different weights expressed by natural numbers (of weight units). Using these weights we are able to weigh out every integer mass up to $1985$ (inclusive). How many ways are there to compose such a set of weight sets given that the joint mass of all weights is the least possible?

Several small villages are situated on the banks of a straight river. On one side, there are 20 villages in a row, and on the other there are 15 villages in a row. We would like to build bridges, each of which connects a village on the one side with a village on the other side. The bridges must not cross, and it should be possible to get from any village to any other village using only those bridges (and not any roads that might exist between villages on the same side of the river). How many different ways are there to build the bridges.

We will say that a set of real numbers $A = (a_1,... , a_{17})$ is stronger than the set of real numbers $B = (b_1, . . . , b_{17})$, and write $A >B$ if among all inequalities $a_i > b_j$ the number of true inequalities is at least $3$ times greater than the number of false. Prove that there is no chain of sets $A_1, A_2, . . . , A_N$ such that $A_1>A_2> \cdots A_N>A_1$. Remark: For 11.4, the constant $3$ is changed to $2$ and $N=3$ and $17$ is changed to $m$ and $n$ in the definition (the number of elements don't have to be equal).

In a summer camp there are $n$ girls $D_1,D_2, ... ,D_n$ and $2n-1$ boys $C_1,C_2, ...,C_{2n-1}$. The girl $D_i, i = 1,2,... ,n,$ knows only the boys $C_1,C_2, ... ,C_{2i-1}$. Let $A(n, r)$ be the number of different ways in which $r$ girls can dance with $r$ boys forming $r$ pairs, each girl with a boy she knows. Prove that $A(n, r) = \binom{n}{r} \frac{r!}{(n-r)!}.$

There are $2000$ cities in the country, each of which has exactly three roads to other cities. Prove that you can close $1000$ roads, so that there is not a single closed route in the country, consisting of an odd number of roads.

It is proposed to partition a set of positive integers into two disjoint subsets $ A$ and $ B$ subject to the conditions [b]i.)[/b] 1 is in $ A$ [b]ii.)[/b] no two distinct members of $ A$ have a sum of the form $ 2^k \plus{} 2, k \equal{} 0,1,2, \ldots;$ and [b]iii.)[/b] no two distinct members of B have a sum of that form. Show that this partitioning can be carried out in unique manner and determine the subsets to which 1987, 1988 and 1989 belong.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal connecting them. Let $d$ be the number of diagonals Alice drew. Find the least possible value that $d$ can obtain.

Consider the following one-person game played on the real line. During the game disks are piled at some of the integer points on the line. To perform a move in the game, the player chooses a point $ j$ at which at least two disks are piled and then takes two disks from the point $ j$ and places one of them at $ j\minus{}1$ and one at $ j\plus{}1$. Initially, $ 2n\plus{}1$ disks are placed at point $ 0$. The player proceeds to perform moves as long as possible. Prove that after $ \frac{1}{6} n(n\plus{}1)(2n\plus{}1)$ moves no further moves will be possible and that at this stage, one disks remains at each of the positions $ \minus{}n,\minus{}n\plus{}1,...,0,...n$.

For which natural numbers $n{}$ it is possible to mark several cells of an $n\times n$ board in such a way that there are an even number of marked cells in all rows and columns, and an odd number on all diagonals whose length is more than one cell? [i]Proposed by S. Berlov[/i]

A game is played on a board with an infinite row of holes labelled $0, 1, 2, \dots$. Initially, $2009$ pebbles are put into hole $1$; the other holes are left empty. Now steps are performed according to the following scheme: (i) At each step, two pebbles are removed from one of the holes (if possible), and one pebble is put into each of the neighbouring holes. (ii) No pebbles are ever removed from hole $0$. (iii) The game ends if there is no hole with a positive label that contains at least two pebbles. Show that the game always terminates, and that the number of pebbles in hole $0$ at the end of the game is independent of the specific sequence of steps. Determine this number.

In an election between $\text{A}$ and $\text{B}$, during the counting of the votes, neither candidate was more than $2$ votes ahead, and the vote ended in a tie, $6$ votes to $6$ votes. Two votes for the same candidate are indistinguishable. In how many orders could the votes have been counted? One possibility is $\text{AABBABBABABA}$.

On a $49\times 69$ rectangle formed by a grid of lattice squares, all $50\cdot 70$ lattice points are colored blue. Two persons play the following game: In each step, a player colors two blue points red, and draws a segment between these two points. (Different segments can intersect in their interior.) Segments are drawn this way until all formerly blue points are colored red. At this moment, the first player directs all segments drawn - i. e., he takes every segment AB, and replaces it either by the vector $\overrightarrow{AB}$, or by the vector $\overrightarrow{BA}$. If the first player succeeds to direct all the segments drawn in such a way that the sum of the resulting vectors is $\overrightarrow{0}$, then he wins; else, the second player wins. Which player has a winning strategy?

Some players participate in a competition. Suppose that each player plays one game against every other player and there is no draw game in the competition. Player $A$ is regarded as an excellent player if the following condition is satisfied: for any other player $B$, either $A$ beats $B$ or there exists another player $C$ such that $C$ beats $B$ and $A$ beats $C$. It is known that there is only one excellent player in the end, prove that this player beats all other players.

Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with $6$ digits( distincts ) and Alex starts the game. Bruno wins if the number with $6$ digits is a prime number, otherwise Alex wins. Which player has the winning strategy?

Each of the cells of an $m \times n$ table is coloured either black or white. For each cell, the total number of the cells which are in the same row or in the same column and of the same colour as this cell is strictly less than the total number of the cells which are in the same row or in the same column and of the other colour as this cell. Prove that in each row and in each column the number of white cells is the same as the number of black ones. (A Shapovalov)

Let $(m,n)$ be pair of positive integers. Julia has carefully planted $m$ rows of $n$ dandelions in an $m \times n$ array in her back garden. Now, Jana un Viviane decides to play a game with a lawnmower they just found. Taking alternating turns and starting with Jana, they can now mow down all the dandelions in a straight horizontal or vertical line (and they must mow down at least one dandelion ). The winner is the player who mows down the final dandelion. Determine all pairs of $(m,n)$ for which Jana has a winning strategy.

A conference has $ 47$ people attending. One woman knows $ 16$ of the men who are attending, another knows $ 17$, and so on up to the $ m$-th woman who knows all $ n$ of the men who are attending. Find $ m$ and $ n$.

We select $16$ cells on an $8\times 8$ chessboard. What is the minimal number of pairs of selected cells in the same row or column?

Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

[b]p1.[/b] Denote $S_n = 1 + \frac12 + \frac13 + ...+ \frac{1}{n}$. What is $144169\cdot S_{144169} - (S_1 + S_2 + ... + S_{144168})$? [b]p2.[/b] Let $A,B,C$ be three collinear points, with $AB = 4$, $BC = 8$, and $AC = 12$. Draw circles with diameters $AB$, $BC$, and $AC$. Find the radius of the two identical circles that will lie tangent to all three circles. [b]p3.[/b] Let $s(i)$ denote the number of $1$’s in the binary representation of $i$. What is $$\sum_{x=1}{314}\left( \sum_{i=0}^{2^{576}-2} x^{s(i)} \right) \,\, mod \,\,629 ?$$ [b]p4.[/b] Parallelogram $ABCD$ has an area of $S$. Let $k = 42$. $E$ is drawn on AB such that $AE =\frac{AB}{k}$ . $F$ is drawn on $CD$ such that $CF = \frac{CD}{k}$ . $G$ is drawn on $BC$ such that $BG = \frac{BC}{k}$ . $H$ is drawn on $AD$ such that $DH = \frac{AD}{k}$ . Line $CE$ intersects $BH$ at $M$, and $DG$ at $N$. Line $AF$ intersects $DG$ at $P$, and $BH$ at $Q$. If $S_1$ is the area of quadrilateral $MNPQ$, find $\frac{S_1}{S}$. [b]p5.[/b] Let $\phi$ be the Euler totient function. What is the sum of all $n$ for which $\frac{n}{\phi(n)}$ is maximal for $1 \le n \le 500$? [b]p6.[/b] Link starts at the top left corner of an $12 \times 12$ grid and wants to reach the bottom right corner. He can only move down or right. A turn is defined a down move immediately followed by a right move, or a right move immediately followed by a down move. Given that he makes exactly $6$ turns, in how many ways can he reach his destination? PS. You had better use hide for answers.

[b]p1.[/b] Let $A(S)$ denote the average value of a set $S$. Let $T$ be the set of all subsets of the set $\{1, 2, 3, 4, ... , 2012\}$, and let $R$ be $\{A(K)|K \in T \}$. Compute $A(R)$. [b]p2.[/b] Consider the minute and hour hands of the Campanile, our clock tower. During one single day ($12:00$ AM - $12:00$ AM), how many times will the minute and hour hands form a right-angle at the center of the clock face? [b]p3.[/b] In a regular deck of $52$ face-down cards, Billy flips $18$ face-up and shuffles them back into the deck. Before giving the deck to Penny, Billy tells her how many cards he has flipped over, and blindfolds her so she can’t see the deck and determine where the face-up cards are. Once Penny is given the deck, it is her job to split the deck into two piles so that both piles contain the same number of face-up cards. Assuming that she knows how to do this, how many cards should be in each pile when he is done? [b]p4.[/b] The roots of the equation $x^3 + ax^2 + bx + c = 0$ are three consecutive integers. Find the maximum value of $\frac{a^2}{b+1}$. [b]p5.[/b] Oski has a bag initially filled with one blue ball and one gold ball. He plays the following game: first, he removes a ball from the bag. If the ball is blue, he will put another blue ball in the bag with probability $\frac{1}{437}$ and a gold ball in the bag the rest of the time. If the ball is gold, he will put another gold ball in the bag with probability $\frac{1}{437}$ and a blue ball in the bag the rest of the time. In both cases, he will put the ball he drew back into the bag. Calculate the expected number of blue balls after $525600$ iterations of this game. [b]p6.[/b] Circles $A$ and $B$ intersect at points $C$ and $D$. Line $AC$ and circle $B$ meet at $E$, line $BD$ and circle $A$ meet at $F$, and lines $EF$ and $AB$ meet at $G$. If $AB = 10$, $EF = 4$, $FG = 8$, find $BG$. PS. You had better use hide for answers.