There are $32$ boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a $15$ day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)

Prove that given $n$ points in the plane, not all aligned, there exists a line that passes through exactly two of them. [hide=original wording]Demostrar que dados n puntos en el plano, no todos alineados, existe una recta que pasa por exactamente dos de ellos.[/hide]

Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given: [list][*]Every row consists of distinct numbers; [*]$\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$; [*]$x_{17}=x_{28}=x_{39}=0$; [*]$x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$. [*]The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$.[/list] Prove that: a) the elements $u_1,u_2,u_3$ come from three different columns; b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists.

Alice creates a sequence: For the first $2025$ terms of this sequence, she writes a random permutation of $\{1;2;3;...;2025\}$. To define the following terms, she does the following: She takes the last $2025$ terms of the sequence, and takes its median. How many values could this sequence's $3000$'th term could get? (Note: To find the median of $2025$ numbers, you write them in an increasing order,and take the number in the middle)

A regular triangle has side length $n{}$. We divided its sides into $n{}$ equal parts and drew a line segment parallel with each side through the dividing points. A lattice of $1+2+\ldots+(n+1)$ intersection points is thus formed. For which positive integers $n{}$ can this lattice be partitioned into triplets of points which are the vertices of a regular triangle of side length $1$? [i]Proposed by Alexander Gunning, Cambridge, UK[/i]

A cube 10x10x10 is constructed from 1000 white unit cubes. Polly and Velly play the following game: Velly chooses a certain amount of parallelepipeds 1x1x10, no two of which have a common vertex or an edge, and repaints them in black. Polly can choose an arbitrary number of unit cubes and ask Velly for their color. What’s the least amount of unit cubes she has to choose so that she can determine the color of each unit cube?

A set of points in the plane is called $\textit{good}$ if the distance between any two points in it is at most $1$. Let $f(n, d)$ be the largest positive integer such that in any $\textit{good}$ set of $3n$ points, there is a circle of diameter $d$, which contains at least $f(n, d)$ points. Prove that there exists a positive real $\epsilon$, such that for all $d \in (1-\epsilon, 1)$, the value of $f(n, d)$ does not depend on $d$ and find that value as a function of $n$.

We consider the graph with vertices $A_1,A_2,\dots A_{2015}$ , $B_1,B_2,\dots B_{2015}$ and edges $A_iA _{i+1}, A_iB_i, B_iB_{i+17} $, taken cyclicaly. Is it true that 4 cops can catch a robber on this graph for every initial position?( First the 4 cops make a move, then the robber makes a move, then the cops make a move etc. A move consists of jumping from the vertex you stay on an adiacent vertex or by staying on your current vertex. Everyone knows the position of everyone everytime. The cops can coordinate their moves. The robber is caught when he shares the same vertex with a cop.) Tuymaada 2017 Q8 Juniors

Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}$ numbers, then Alberto erases $2^{8}$ numbers, then Barbara $2^{7}$ and so on, until there are only two numbers a,b left. Now Barbara earns $|a-b|$ euro. Find the maximum number of euro that Barbara can always win, independently of Alberto's strategy.

Players Aino and Eino take turns choosing numbers from the set $\{0,..., n\}$ with $n\in \Bbb{N}$ being fixed in advance. The game ends when the numbers picked by one of the players include an arithmetic progression of length $4.$ The one who obtains the progression wins. Prove that for some $n,$ the starter of the game wins. Find the smallest such $n.$

Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different signs. At the beginning they are given $20$ cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is $15$. At the end of the game, one player has a $5$ and a $3$ in his hand, on the table there's a $9$, the other player has a card in his hand. What is it's value?

Let $M, k>0$ integers. Let $X(M,k)$ the (infinite) set of all integers that can be factored as ${p_1}^{e_1} \cdot {p_2}^{e_2} \cdot \ldots \cdot {p_r}^{e_r}$ where each $p_i$ is not smaller than $M$ and also each $e_i$ is not smaller than $k$. Let $Z(M,k,n)$ the number of elements of $X(M,k)$ not bigger than $n$. Show that there are positive reals $c(M,k)$ and $\beta(M,k)$ such that $$\lim_{n \rightarrow \infty}{\frac{Z(M,k,n)}{n^{\beta(M,k)}}} = c(M,k)$$ and find $\beta(M,k)$

We have $2021$ colors and $2021$ chips of each color. We place the $2021^2$ chips in a row. We say that a chip $F$ is [i]bad[/i] if there is an odd number of chips that have a different color to $F$ both to the left and to the right of $F$. (a) Determine the minimum possible number of bad chips. (b) If we impose the additional condition that each chip must have at least one adjacent chip of the same color, determine the minimum possible number of bad chips.

On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place?

It is known that there are $2024$ pairs of friends among $100$ people. Show that is possible to split them into $50$ pairs so that: (a) There are at most $20$ pairs that are friends with each other; (b) There are at least $23$ pairs that are friends with each other; (c) There are exactly $22$ pairs that are friends with each other.

Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$

There's a group of $21$ people such that each person has no more than $7$ friends among the others and any two friends have a different number of total friends. Prove that there are $6$ people, none of which knows the others.

A simple graph is called "divisibility", if it's possible to put distinct numbers on its vertices such that there is an edge between two vertices if and only if number of one of its vertices is divisible by another one. A simple graph is called "permutationary", if it's possible to put numbers $1,2,...,n$ on its vertices and there is a permutation $ \pi $ such that there is an edge between vertices $i,j$ if and only if $i>j$ and $\pi(i)< \pi(j)$ (it's not directed!) Prove that a simple graph is permutationary if and only if its complement and itself are divisibility. [i]Proposed by Morteza Saghafian[/i] .

Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$. (Peter Cameron)

Let $n$ be a positive integer. Mary writes the $n^3$ triples of not necessarily distinct integers, each between $1$ and $n$ inclusive on a board. Afterwards, she finds the greatest (possibly more than one), and erases the rest. For example, in the triple $(1, 3, 4)$ she erases the numbers 1 and 3, and in the triple $(1, 2, 2)$ she erases only the number 1, Show after finishing this process, the amount of remaining numbers on the board cannot be a perfect square.

positive real $C$ is $n-vengeful$ if it is possible to color the cells of an $n \times n$ chessboard such that: i) There is an equal number of cells of each color. ii) In each row or column, at least $Cn$ cells have the same color. a) Show that $\frac{3}{4}$ is $n-vengeful$ for infinitely many values of $n$. b) Show that it does not exist $n$ such that $\frac{4}{5}$ is $n-vengeful$.

In a far away kingdom, there exist $k^2$ cities subdivided into k distinct districts, such that in the $i^ {th}$ district, there exist $2i - 1$ cities. Each city is connected to every city in its district but no cities outside of its district. In order to improve transportation, the king wants to add $k - 1$ roads such that all cities will become connected, but his advisors tell him there are many ways to do this. Two plans are different if one road is in one plan that is not in the other. Find the total number of possible plans in terms of $k$.

Without lifting pen from paper, we draw a polygon in such away that from every two adjacent sides one of them is vertical. In addition, while drawing the polygon all vertical sides have been drawn from up to down. Prove that this polygon has cut itself.

Nicu plays the Next game on the computer. Initially the number $S$ in the computer has the value $S = 0$. At each step Nicu chooses a certain number $a$ ($0 <a <1$) and enters it in computer. The computer arbitrarily either adds this number $a$ to the number $S$ or it subtracts from $S$ and displays on the screen the new result for $S$. After that Nicu does Next step. It is known that among any $100$ consecutive operations the computer the at least once apply the assembly. Give an arbitrary number $M> 0$. Show that there is a strategy for Nicu that will always allow him after a finite number of steps to get a result $S> M$. [hide=original wording]Nicu joacă la calculator următorul joc. Iniţial numărul S din calculator are valoarea S = 0. La fiecare pas Nicu alege un număr oarecare a (0 < a < 1) şi îl introduce în calculator. Calculatorul, în mod arbitrar, sau adună acest număr a la numărul S sau îl scade din S şi afişează pe ecran rezultatul nou pentru S. După aceasta Nicu face următorul pas. Se ştie că printre oricare 100 de operaţii consecutive calculatorul cel puţin o dată aplică adunarea. Fie dat un număr arbitrar M > 0. Să se arate că există o strategie pentru Nicu care oricând îi va permite lui după un număr finit de paşi să obţină un rezulat S > M.[/hide]