Five players $(A,B,C,D,E)$ take part in a bridge tournament. Every two players must play (as partners) against every other two players. Any two given players can be partners not more than once per a day. What is the least number of days needed for this tournament?

The shapes in the image consist of six unit cubes. Which of the following 3D objects can be filled up with the aforementioned shapes: a) a cube with side length $3$, from which one edge has been removed (i.e. three layers of the shape [img]https://i.imgur.com/vUqgHS2.png[/img] )? b) a rectangular prism of size $5 \times 4 \times 3$, from which two edges of length $3$ have been removed from one of the $5 \times 3$ sides (i.e. three layers of the shape [img]https://imgur.com/W4pfEfz.png[/img] )? We can use each of shapes at most once, no two shapes can overlap, nor protrude from the 3D object and every unit cube of the 3D object must be covered by a unit cube of one of the constituent shapes. [center][img]https://imgur.com/evAmuep.png[/img][/center] [i]Proposed by Ilija Jovčeski[/i]

A [i]uniform covering[/i] of the integers $1,2,...,n$ is a finite multiset of subsets of $\{1,2,...,n\}$, so that each number lies in the same amount of sets from the covering. A covering may contain the same subset multiple times, it must contain at least one subset, and it may contain the empty subset. For example, $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})$ is a uniform covering of $1,2,3,4$ (every number occurs in two sets). The covering containing only the empty set is also uniform (every number occurs in zero sets). Given two uniform coverings, we define a new uniform covering, their [i]sum[/i] (denoted by $\oplus$), by adding the sets from both coverings. For example: $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})\oplus(\{1\},\{2\},\{3\},\{4\})=$ $(\{1\},\{1\},\{1\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\})$ A uniform covering is called [i]non-composite[/i] if it's not a sum of two uniform coverings. Prove that for any $n\geq1$, there are only finitely many non-composite uniform coverings of $1,2,...,n$.

[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. [b](b)[/b]What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

There are $n$ cards on a table in a line, with a positive real written on eachcard. LTF and Sunny are playing a game where they take turns taking away the first or the last card in line. The player that has the bigger sum of all the numberson his cards wins. If LTF goes first, find all $n$ such that LTF can always prevent Sunny from winning, regardless of the numbers written on the cards.

In each square of a $2012\times 2012$ board there's a person. People are either honest, who always tell the truth, or liars, who always lie. At a given moment, each person makes the same statement: "In my row there are the same number of liars as in my column." Determine the minimum number of honest people that can be on the board.

There are $75$ points in the plane, no three collinear. Prove that the number of acute triangles is no more than $70\%$ from the total number of triangles with vertices in these points.

How many distinct arrangements are possible for wearing five different rings in the five fingers of the right hand? (We can wear multiple rings in one finger) [list=1] [*] $\frac{10!}{5!}$ [*] $5^5$ [*] $\frac{9!}{4!}$ [*] None of these [/list]

A square board with dimensions $100 \times 100$ is divided into $10 000 $unit squares. One of the squares is cut out. Is it possible to cover the rest of the board by isosceles right angled triangles which have hypotenuses of length $2$, and in such a way that their hypotenuses lie on sides of the squares and their other two sides lie on diagonals? The triangles must not overlap each other or extend beyond the edges of the board. (S Fomin, Leningrad)

We put a figure of a king on some $6 \times 6$ chessboard. It can in one thrust jump either vertically or horizontally. The length of this jump is alternately one and two squares, whereby a jump of one (i.e. to the adjacent square) of the piece begins. Decide whether you can choose the starting position of the pieces so that after a suitable sequence $35$ jumps visited each box of the chessboard just once.

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

Let $E$ be a family of subsets of $\{1,2,\ldots,n\}$ with the property that for each $A\subset \{1,2,\ldots,n\}$ there exist $B\in F$ such that $\frac{n-d}2\leq |A \bigtriangleup B| \leq \frac{n+d}2$. (where $A \bigtriangleup B = (A\setminus B) \cup (B\setminus A)$ is the symmetric difference). Denote by $f(n,d)$ the minimum cardinality of such a family. a) Prove that if $n$ is even then $f(n,0)\leq n$. b) Prove that if $n-d$ is even then $f(n,d)\leq \lceil \frac n{d+1}\rceil$. c) Prove that if $n$ is even then $f(n,0) = n$

The numbers from $1$ to $50$ are printed on cards. The cards are shuffled and then laid out face up in $5$ rows of $10$ cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrangement, do the cards in the rows still increase from left to right?

Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn. $A$ starts the game. For each positive integer $n$, find a winning strategy for one of the players.

How many colorings of an $n$-gon in $p \ge 2$ colors are there such that no two neighboring vertices have the same color?

For each positive real number $\alpha$, define $$\lfloor \alpha \mathbb{N}\rfloor :=\{\lfloor \alpha m \rfloor\; |\; m\in \mathbb{N}\}.$$ Let $n$ be a positive integer. A set $S\subseteq \{1,2,\ldots,n\}$ has the property that: for each real $\beta >0$, $$ \text{if}\; S\subseteq \lfloor \beta \mathbb{N} \rfloor, \text{then}\; \{1,2,\ldots,n\} \subseteq \lfloor \beta\mathbb{N}\rfloor.$$ Determine, with proof, the smallest positive size of $S$.

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

There are $n$ aborigines on an island. Any two of them are either friends or enemies. One day, the chieftain orders that all citizens (including himself) make and wear a necklace with zero or more stones so that: (i) given a pair of friends, there exists a color such that each has a stone of that color; (ii) given a pair of enemies,there does not exist a color such that each a stone of that color. (a) Prove that the aborigines can carry out the chieftain’s order. (b) What is the minimum number of colors of stones required for the aborigines to carry out the chieftain’s order?

Place a pebble at each [i]non-positive[/i] integer point on the real line, and let $n$ be a fixed positive integer. At each step we choose some n consecutive integer points, remove one of the pebbles located at these points and rearrange all others arbitrarily within these points (placing at most one pebble at each point). Determine whether there exists a positive integer $n$ such that for any given $N > 0$ we can place a pebble at a point with coordinate greater than $N$ in a finite number of steps described above.

Ten students take a test consisting of $4$ different papers in Algebra, Geometry, Number Theory and Combinatorics. First, the proctor distributes randomly the Algebra paper to each student. Then the remaining papers are distributed one at a time in the following order: Geometry, Number Theory, Combinatorics in such a way that no student receives a paper before he finishes the previous one. In how many ways can the proctor distribute the test papers given that a student may for example nish the Number Theory paper before another student receives the Geometry paper, and that he receives the Combinatorics paper after that the same other student receives the Combinatorics papers.

Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$. Note: The tiles must completely cover all the board, with no overlappings.

Rachel has two indistinguishable tokens, and places them on the first and second square of a $1 \times 6$ grid of squares, She can move the pieces in two ways: $\bullet$ If a token has free square in front of it, then she can move this token one square to the right. $\bullet$ If the square immediately to the right of a token is occupied by the other token, then she can “leapfrog” the first token; she moves the first token two squares to the right, over the other token, so that it is on the square immediately to the right of the other token. If a token reaches the $6$th square, then it cannot move forward any more, and Rachel must move the other one until it reaches the $5$th square. How many different sequences of moves for the tokens can Rachel make so that the two tokens end up on the $5$th square and the 6th square?

A [u]positive sequence[/u] is a finite sequence of positive integers. [u]Sum of a sequence[/u] is the sum of all the elements in the sequence. We say that a sequence $A$ can be [u]embedded[/u] into another sequence $B$, if there exists a strictly increasing function $$\phi : \{1,2, \ldots, |A|\} \rightarrow \{1,2, \ldots, |B|\},$$ such that $\forall i \in \{1, 2, \ldots ,|A|\}$, $$A[i] \leq B[\phi(i)],$$ where $|S|$ denotes the length of a sequence $S$. For example, $(1,1,2)$ can be embedded in $(1,2,3)$, but $(3,2,1)$ can not be in $(1,2,3)$\\ Given a positive integer $n$, construct a positive sequence $U$ with sum $O(n \, \log \, n)$, such that all the positive sequences with sum $n$, can be embedded into $U$.\\

Let $\mathcal{S}$ be a set of $2023$ points in a plane, and it is known that the distances of any two different points in $S$ are all distinct. Ivan colors the points with $k$ colors such that for every point $P \in \mathcal{S}$, the closest and the furthest point from $P$ in $\mathcal{S}$ also have the same color as $P$. What is the maximum possible value of $k$? [i]Proposed by Ivan Chan Kai Chin[/i]

$11$ people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and $11$ cards with numbers $1$ to $11$ are given to them. Some may have no card and some may have more than $1$ card. In each round, one [and only one] can give one of his cards with number $ i $ to his adjacent person if after and before the round, the locations of the cards with numbers $ i-1,i,i+1 $ don’t make an acute-angled triangle. (Card with number $0$ means the card with number $11$ and card with number $12$ means the card with number $1$!) Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.) Prove that the cards can’t be gathered at one person.