In a certain country there are $n$ cities. Some pairs of cities are connected by highways in such a way that for each two cities there is at most one highway connecting them. Assume that for a certain positive integer $k$, the total number of highways is greater than $\frac{nk}{2}$. Show that there exist $k+2$ distinct cities $C_1, C_2, \dots, C_{k+2}$ such that $C_i$ and $C_{i+1}$ are connected by a highway for $i=1, 2, \dots, k+1$. [i]Proposed by Oriol Solé[/i]

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

Let $X$ be the set of all lattice points in the plane (points $(x, y)$ with $x, y \in \mathbb{Z}$). A path of length $n$ is a chain $(P_0, P_1, ... , P_n)$ of points in $X$ such that $P_{i-1}P_i = 1$ for $i = 1, ... , n$. Let $F(n)$ be the number of distinct paths beginning in $P_0=(0,0)$ and ending in any point $P_n$ on line $y = 0$. Prove that $F(n) = \binom{2n}{n}$

$11$ students take a test. For any two question in a test, there are at least $6$ students who solved exactly one of those two questions. Prove that there are no more than $12$ questions in this test. Showing the equality case is not needed.

Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck, the cards that were in the deck of size \(n\) appear in order, as do the cards that were in the deck of size \(52-n\).) Alice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee? [i]Proposed by Espen Slettnes[/i]

Two treasure-hunters found a treasure containing coins of value $a_1< a_2 < ... < a_{2003}$ (the quantity of coins of each value is unlimited). The first treasure-hunter forms all the possible sets of different coins containing odd number of elements, and takes the most valuable coin of each such set. The second treasure-hunter forms all the possible sets of different coins containing even number of elements, and takes the most valuable coin of each such set. Which one of them is going to have more money and how much more? (H. Nestra)

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 $n\ge 2$ be a positive integer, and consider an $n\times n$ grid of $n^2$ equilateral triangles. Two triangles are adjacent if they share at least one vertex. Each triangle is colored red or blue, splitting the grid into regions. Find, with proof, the minimum number of triangles in the largest region. [i]Rohan Bodke[/i]

We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated. It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?

In connection with the formation of a stable government, the President invited all $240$ Members of Parliament to three separate consultations, where each member participated in exactly one consultation, and at each consultation there has been at least one member present. Discussions between pairs of members are to take place to discuss the consultations. Is it possible for these discussions to occur in such a way that there exists a non-negative integer $k$, such that for every two members who participated in different consultations, there are exactly $k$ members who participated in the remaining consultation, with whom each of the two members has a conversation, and exactly $k$ members who participated in the remaining consultation, with whom neither of the two has a conversation? If yes, then find all possible values of $k$.

Let $S = \left\{ 1, \dots, 100 \right\}$, and for every positive integer $n$ define \[ T_n = \left\{ (a_1, \dots, a_n) \in S^n \mid a_1 + \dots + a_n \equiv 0 \pmod{100} \right\}. \] Determine which $n$ have the following property: if we color any $75$ elements of $S$ red, then at least half of the $n$-tuples in $T_n$ have an even number of coordinates with red elements. [i]Ray Li[/i]

$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

A positive integer $m(\ge 2$) is given. From circle $C_1$ with a radius 1, construct $C_2, C_3, C_4, ... $ through following acts: In the $i$th act, select a circle $P_i$ inside $C_i$ with a area $\frac{1}{m}$ of $C_i$. If such circle dosen't exist, the act ends. If not, let $C_{i+1}$ a difference of sets $C_i -P_i$. Prove that this act ends within a finite number of times.

All vertices of a regular 100-gon are colored in 10 colors. Prove that there exist 4 vertices of the given 100-gon which are the vertices of a rectangle and which are colored in at most 2 colors.

A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] [i](T. Doslic)[/i]

Given $2021$ distinct positive integers non divisible by $2^{1010}$, show that it's always possible to choose $3$ of them $a$, $b$ and $c$, such that $|b^2-4ac|$ is not a perfect square.

Given a billiard in the form of a regular $1998$-gon $A_1A_2...A_{1998}$. A ball was released from the midpoint of side $A_1A_2$, which, reflected therefore from sides $A_2A_3$, $A_3A_4$, . . . , $A_{1998}A_1$ (according to the law, the angle of incidence is equal to the angle of reflection), returned to the starting point. Prove that the trajectory of the ball is a regular $1998$-gon.

The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$. Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums $A\le 8$ and $B\le 4$. [i](I. Gorodnin)[/i]

Given a set of 1988 points in the plane. No four points of the set are collinear. The points of a subset with 1788 points are coloured blue, the remaining 200 are coloured red. Prove that there exists a line in the plane such that each of the two parts into which the line divides the plane contains 894 blue points and 100 red points.

The two figures depicted below consisting of $6$ and $10$ unit squares, respectively, are called staircases. Consider a $2018\times 2018$ board consisting of $2018^2$ cells, each being a unit square. Two arbitrary cells were removed from the same row of the board. Prove that the rest of the board cannot be cut (along the cell borders) into staircases (possibly rotated).

A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$, the smallest natural $k$ that is $n$-squared.

Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.) What is the minimal value of $N$ for which this is possible?

Let $k$ be a positive integer. Scrooge McDuck owns $k$ gold coins. He also owns infinitely many boxes $B_1, B_2, B_3, \ldots$ Initially, bow $B_1$ contains one coin, and the $k-1$ other coins are on McDuck's table, outside of every box. Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes: - if two consecutive boxes $B_i$ and $B_{i+1}$ both contain a coin, McDuck can remove the coin contained in box $B_{i+1}$ and put it on his table; - if a box $B_i$ contains a coin, the box $B_{i+1}$ is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box $B_{i+1}$. As a function of $k$, which are the integers $n$ for which Scrooge McDuck can put a coin in box $B_n$?

Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules: i) Any incantation can appear no more than once; ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it; iii)The first person who cannot spell an incantation loses the contest. Answer the following questions: a) If A says '$STAGEPREIMO$' first, then who will win? b) Let $M$ be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are $2009$ and containing only four letters $A,B,C,D$, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in $M$ such that A has a winning strategy by starting with it.

At the faculty of mathematics study $40$ boys and $10$ girls. Every girl acquaintance with all boys, who older than her, or tall(higher) than her. Prove that there exist two boys such that the sets of acquainted-girls of the boys are same.