In an $n$ by $n$ grid, each cell is filled with an integer between $1$ and $6$. The outmost cells all contain the number $1$, and any two cells that share a vertex has difference not equal to $3$. For any vertex $P$ inside the grid (not including the boundary), there are $4$ cells that have $P$ has a vertex. If these four cells have exactly three distinct numbers $i$, $j$, $k$ (two cells have the same number), and the two cells with the same number have a common side, we call $P$ an $ijk$-type vertex. Let there be $A_{ijk}$ vertices that are $ijk$-type. Prove that $A_{123}\equiv A_{246} \pmod 2$.

Determine the largest $k$ such that for all competitive graph with $2019$ points, if the difference between in-degree and out-degree of any point is less than or equal to $k$, then this graph is strongly connected.

Rodolfo and Gabriela have $9$ chips numbered from $1$ to $9$ and they have fun with the following game: They remove the chips one by one and alternately (until they have $3$ chips each), with the following rules: $\bullet$ Rodolfo begins the game, choosing a chip and in the following moves he must remove, each time, a chip three units greater than the last chip drawn by Gabriela. $\bullet$ Gabriela, on her turn, chooses a first chip and in the following times she must draw, each time, a chip two units smaller than the last chip that she herself drew. $\bullet$ The game is won by whoever gets the highest number by adding up their three tokens. $\bullet$ If the game cannot be completed, a tie is declared. If they play without making mistakes, how should Rodolfo play to be sure he doesn't lose?

The European zoos with exactly $100$ types of species each are separated into two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A, B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. Prove that we can colour the cages in $3$ colours (all animals of the same type live in the same cage) such that no zoo has cages of only one colour

Q.A bug travels in the co-ordinate plane moving along only the lines that are parallel to the $X$ and $Y$ axes.Let $A=(-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$.How many lattice points lie on at least one of these paths. My answer ($87$)

There are $33$ children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers $0,1,2,\dots,10$ appear at least once on the blackboard. Prove that there are at least two children in this class that have the same forename and surname.

The numbers $1$ to $12$ are arranged in a sequence. The number of ways this can be done equals $12 \times11 \times 10\times ...\times 1$. We impose the condition that in the sequence there should be exactly one number that is smaller than the number directly preceding it. How many of the $12 \times11 \times 10\times ...\times 1$ sequences satisfy this condition?

Find the number of ways one can put numbers $1$ or $2$ in each cell of an $8\times 8$ chessboard in such a way that the sum of the numbers in each column and in each row is an odd number. (Two ways are considered different if the number in some cell in the first way is different from the number in the cell situated in the corresponding position in the second way)

Five points are chosen uniformly at random on a segment of length $1$. What is the expected distance between the closest pair of points?

The sum of several positive integers, not necessarily different, all of them less than or equal to $10$, is equal to $S$. We want to distribute all these numbers into two groups such that the sum of the numbers in each group is less than or equal to $80.$ Determine all values ​​of $S$ for which this is possible.

Prove that the number of incidences of $n$ distinct points on $n$ distinct lines in plane is $\mathcal O (n^{\frac{4}{3}})$. Find a configuration for which $\Omega (n^{\frac{4}{3}})$ incidences happens.

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

A regular polygon of $ n$ sides ($ n\geq3$) has its vertex numbered from 1 to $ n$. One draws all the diagonals of the polygon. Show that if $ n$ is odd, it is possible to assign to each side and to each diagonal an integer number between 1 and $ n$, such that the next two conditions are simultaneously satisfied: (a) The number assigned to each side or diagonal is different to the number assigned to any of the vertices that is endpoint of it. (b) For each vertex, all the sides and diagonals that have it as an endpoint, have different number assigned.

An infinite number of squares on an infinitely square grid paper are painted red. Show that you can draw a number of squares on the paper, with sides along the grid lines, such that: (1) no square in the grid belongs to more than one square (an edge, on the other hand, may belong to more than one square) (2) each red square is located in one of the squares and the number of red squares in such square is at least $1/5$ and at most $4/5$ of the number of squares in the square. [hide=original wording] Ett andligt antal rutor pa ett oandligt rutat papper ar malade roda. Visa att man pa papperet kan rita in ett antal kvadrater, med sidor utefter rutnatets linjer, sadana att : (1) ingen ruta i natet tillhor mer an en kvadrat (en kant kan daremot tillhora mer an en kvadrat), (2) varje rod ruta ligger i nagon av kvadraterna och antalet roda rutor i en sadan kvadratar minst 1/5 och hogst 4/5 av antalet rutor i kvadraten. [url=http://www.mattetavling.se/wp-content/uploads/2011/01/Final10.pdf]source[/url][/hide] PS. I always post the original wording when I doubt about my (using Google) translation.

If $a$ and $b$ are selected uniformly from $\{0,1,\ldots,511\}$ without replacement, the expected number of $1$'s in the binary representation of $a+b$ can be written in simplest from as $\tfrac{m}{n}$. Compute $m+n$.

p1. Find an integer that has the following properties: a) Every two adjacent digits in the number are prime. b) All prime numbers referred to in item (a) above are different. p2. Determine all integers up to $\sqrt{50+\sqrt{n}}+\sqrt{50-\sqrt{n}}$ p3. The following figure shows the path to form a series of letters and numbers “OSN2015”. Determine as many different paths as possible to form the series of letters and numbers by following the arrows. [img]https://cdn.artofproblemsolving.com/attachments/6/b/490a751457871184a506c2966f8355f20cebbd.png[/img] p4. Given an acute triangle $ABC$ with $L$ as the circumcircle. From point $A$, a perpendicular line is drawn on the line segment $BC$ so that it intersects the circle $L$ at point $X$. In a similar way, a perpendicular line is made from point $B$ and point $C$ so that it intersects the circle $L$, at point $Y$ and point $Z$, respectively. Is arc length $AY$ = arc length $AZ$ ? p5. The students of class VII.3 were divided into five groups: $A, B, C, D$ and $E$. Each group conducted five science experiments for five weeks. Each week each group performs an experiment that is different from the experiments conducted by other groups. Determine at least two possible trial schedules in week five, based on the following information: $\bullet$ In the first week, group$ D$ did experiment $4$. $\bullet$ In the second week, group $C$ did the experiment $5$. $\bullet$ In the third week, group $E$ did the experiment $5$. $\bullet$ In the fourth week, group $A$ did experiment $4$ and group $D$ did experiment $2$.

There are some cities in both sides of a river and there are some sailing channels between the cities. Each sailing channel connects exactly one city from a side of the river to a city on the other side. Each city has exactly $k$ sailing channels. For every two cities, there's a way which connects them together. Prove that if we remove any (just one) sailing channel, then again for every two cities, there's a way that connect them together. $( k \geq 2)$

There are $30$ students in a class. In an examination, their results were all different from each other. It is given that everyone has the same number of friends. Find the maximum number of students such that each one of them has a better result than the majority of his friends. PS. Here majority means larger than half.

Fill up each square of a $50$ by $50$ grid with an integer. Let $G$ be the configuration of $8$ squares obtained by taking a $3$ by $3$ grid and removing the central square. Given that for any such $G$ in the $50$ by $50$ grid, the sum of integers in its squares is positive, show there exist a $2$ by $2$ square such that the sum of its entries is also positive.

There are six empty purses on the table. How many ways are there to put 12 two-euro coins in purses in such a way that at most one purse remains empty?

Let $T_n$ be the number of non-congruent triangles with positive area and integer side lengths summing to $n$. Prove that $T_{2022}=T_{2019}$. [i]Proposed by Constantinos Papachristoforou[/i]

From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1, \cdots, q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that : [b](i)[/b] each pair of consecutive teams on the list have one member in common and [b](ii)[/b] the chain of teams on the list are in rank order. [i]Alternative formulation.[/i] Given a graph with $n$ vertices and $q$ edges numbered $1, \cdots , q$, show that there exists a chain of $m$ edges, $m \geq \frac{2q}{n}$ , each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Real numbers are placed at the vertices of an $n{}$-gon. On each side, we write the sum of the numbers on its endpoints. For which $n{}$ is it possible that the numbers on the sides form a permutation of $1, 2, 3,\ldots , n$? [i]From the folklore[/i]

In the city of Flensburg there is a single, infinitely long, street with housesnumbered $2, 3, \ldots$. The police in Flensburg is trying to catch a thief who every night moves from the house where she is currently hiding to one of its neighbouring houses. To taunt the local law enforcement the thief reveals every morning the highest prime divisor of the number of the house she has moved to. Every Sunday afternoon the police searches a single house, and they catch the thief if they search the house she is currently occupying. Does the police have a strategy to catch the thief in finite time?