Let $n$ and $d$ be positive integers satisfying $d<\dfrac{n}{2}$. There are $n$ boys and $n$ girls in a school. Each boy has at most $d$ girlfriends and each girl has at most $d$ boyfriends. Prove that one can introduce some of them to make each boy have exactly $2d$ girlfriends and each girl have exactly $2d$ boyfriends. (I think we assume if a girl has a boyfriend, she is his girlfriend as well and vice versa) (proposed by B. Batbaysgalan, folklore).

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

Alice and Bob play a game on a connected graph with $2n$ vertices, where $n\in \mathbb{N}$ and $n>1$.. Alice and Bob have tokens named A and B respectively. They alternate their turns with Alice going first. Alice gets to decide the starting positions of A and B. Every move, the player with the turn moves their token to an adjacent vertex. Bob's goal is to catch Alice, and Alice's goal is to prevent this. Note that positions of A, B are visible to both Alice and Bob at every moment. Provided they both play optimally, what is the maximum possible number of edges in the graph if Alice is able to evade Bob indefinitely? [i]Proposed by Shashank Ingalagavi and Vighnesh Sangle[/i]

Prove that there are infinitely many n’s for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$, and prove that there are infinitely many $n$’s for which there is no such partition.

Consider a convex polygon in a plane such that the length of all edges and diagonals are rational. After connecting all diagonals, prove that any length of a segment is rational.

On a blackboard there are $2010$ natural nonzero numbers. We define a "move" by erasing $x$ and $y$ with $y\neq0$ and replacing them with $2x+1$ and $y-1$, or we can choose to replace them by $2x+1$ and $\frac{y-1}{4}$ if $y-1$ is divisible by 4. Knowing that in the beginning the numbers $2006$ and $2008$ have been erased, show that the original set of numbers cannot be attained again by any sequence of moves.

A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. The domino may be turned around. There is one domino of each type, so $28$ in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino $(3,4)$, total $3 + 4 = 7$. Then $(1,3)$, total $1 + 3 + 3 + 4 = 11$, then $(4,4)$, total $11 + 4 + 4 = 19$. What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?

Let $S$ be a finite set and $P$ the set of all subsets of $S$. Show that one can label the elements of $P$ as $A_i$ such that (1) $A_1 =\emptyset$. (2) For each $n\geq1 $ we either have $A_{n-1}\subset A_{n}$ and $|A_{n} \setminus A_{n-1}|=1$ or $A_{n}\subset A_{n-1}$ and $|A_{n-1} \setminus A_{n}|=1.$

An integer $ n$ is called good if there are $ n \geq 3$ lattice points $ P_1, P_2, \ldots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $ P_iP_j$ has a rational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have irrational lengths; and if line segment $ P_iP_j$ has an irrational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have rational lengths. (1) Determine the minimum good number. (2) Determine if 2005 is a good number. (A point in the coordinate plane is a lattice point if both of its coordinate are integers.)

There are five dots arranged in a line from left to right. Each of the dots is colored from one of five colors so that no $3$ consecutive dots are all the same color. How many ways are there to color the dots?

After the schoolday is over, Juku must attend an extra math class. The teacher writes a quadratic equation $ x^2\plus{} p_1x\plus{}q_1 \equal{} 0$ with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation $ x^2 \plus{} p_2x \plus{} q_2 \equal{} 0,$ where $ p_2$ and $ q_2$ are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for $ p_1$ and $ q_1$ such that the teacher can hold Juku at school forever.

There are $2^n$ words of length $n$ over the alphabet $\{0, 1\}$. Prove that the following algorithm generates the sequence $w_0, w_1, \ldots, w_{2^n-1}$ of all these words such that any two consecutive words differ in exactly one digit. (1) $w_0 = 00 \ldots 0$ ($n$ zeros). (2) Suppose $w_{m-1} = a_1a_2 \ldots a_n,\quad a_i \in \{0, 1\}$. Let $e(m)$ be the exponent of $2$ in the representation of $n$ as a product of primes, and let $j = 1 + e(m)$. Replace the digit $a_j$ in the word $w_{m-1}$ by $1 - a_j$. The obtained word is $w_m$.

$3n$ lines are drawn on the plane ($n > 1$), such that no two of them are parallel and no three of them are concurrent. Prove that, if $2n$ of the lines are coloured red and the other $n$ lines blue, there are at least two regions of the plane such that all of their borders are red. Note: for each region, all of its borders are contained in the original set of lines, and no line passes through the region.

In a certain country there are $10$ cities connected by a network of one-way nonstop flights so that it is possible to fly (using one or more flights) from any city to any other. Let $n$ be the least number of flights needed to complete a trip starting from one of the cities, visiting all others and returning to the starting point. Find the greatest possible value of $n$.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

Farmer James invents a new currency, such that for every positive integer $n\le 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be [i]nice[/i] if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?

At Easter Day, $4$ children and their mothers participated in a game in which they had to find hidden chocolate eggs. Augustine found $4$ eggs, Bruno found $6$, Carlos found $9$ and Daniel found $12$. Mrs. Gómez found the same number of eggs as her son, Mrs. Junco found twice as many eggs as her son, Mrs. Messi found three times as many eggs as her son, and Mrs. Núñez found five times as many eggs as her son. At the end of the day, they put all the eggs in boxes, with $18$ eggs in each box, and only one egg was left over. Determine who the mother of each child is.

Given is an integer $n \geq 1$ and an $n \times n$ board, whose all cells are initially white. Peter the painter walks around the board and recolors the visited cells according to the following rules. Each walk of Peter starts at the bottom-left corner of the board and continues as follows: - if he is standing on a white cell, he paints it black and moves one cell up (or walks off the board if he is in the top row); - if he is standing on a black cell, he paints it white and moves one cell to the right (or walks off the board if he is in the rightmost column). Peter’s walk ends once he walks off the board. Determine the minimum positive integer $s$ with the following property: after exactly $s$ walks all the cells of the board will become white again.

In the Flatland Congress there are senators who are on committees. Each senator is on at least one committee, and each committee has at least one senator. The rules for forming committees are as follows: $\bullet$ For any pair of senators, there is exactly one committee which contains both senators. $\bullet$ For any two committees, there is exactly one senator who is on both committees. $\bullet$ There exist a set of four senators, no three of whom are all on the same committee. $\bullet$ There exists a committee with exactly $6$ senators. If there are at least $25$ senators in this Congress, compute the minimum possible number of senators $s$ and minimum number of committees $c$ in this Congress. Express your answer in the form $(s, c)$.

On the Cartesian plane, all points with both integer coordinates are painted blue. Two blue points are said to be [i]mutually visible[/i] if the line segment that connects them has no other blue points . Prove that there is a set of $ 2019$ blue points that are mutually visible two by two. [hide=official wording]No plano cartesiano, todos os pontos com ambas coordenadas inteiras são pintados de azul. Dois pontos azuis são ditos mutuamente visíveis se o segmento de reta que os conecta não possui outros pontos azuis. Prove que existe um conjunto de 2019 pontos azuis que são mutuamente visíveis dois a dois.[/hide] PS. There is a comment about problem being wrong / incorrect [url=https://artofproblemsolving.com/community/c6h1957974p14780265]here[/url]

From the $9 \times 9$ chessboard, all $16$ unit squares whose row numbers and column numbers are both even have been removed. Disect the punctured board into rectangular pieces, with as few of them being unit squares as possible.

Each cell of a $100\times 100$ grid is colored with one of $101$ colors. A cell is [i]diverse[/i] if, among the $199$ cells in its row or column, every color appears at least once. Determine the maximum possible number of diverse cells.

There are given $2^{500}$ points on a circle labeled $1,2,\ldots ,2^{500}$ in some order. Prove that one can choose $100$ pairwise disjoint chords joining some of theses points so that the $100$ sums of the pairs of numbers at the endpoints of the chosen chord are equal.

A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice [b]nice[/b] if at the end nobody has her own ball and it is called [b]tiresome[/b] if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.

Javiera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next Javiera says a natural number $k$, $1 \le k \le n$ and Claudio must move the token to the right or to the left at your choice $k$ squares and so on. Javiera wins if she manages to remove the piece that Claudio moves from the board. Determine the smallest $n$ such that Javiera always wins after a finite number of moves.