Each of the numbers $1, 2, 3,... , 2006^2$ is placed at random into a cell of a $2006\times 2006$ board. Prove that there exist two cells which share a common side or a common vertex such that the sum of the numbers in them is divisible by $4$. (4)

Six teams participate in a hockey tournament. Each team plays exactly once against each other team. A team is awarded $3$ points for each game they win, $1$ point for each draw, and $0$ points for each game they lose. After the tournament, a ranking is made. There are no ties in the list. Moreover, it turns out that each team (except the very last team) has exactly $2$ points more than the team ranking one place lower. Prove that the team that finished fourth won exactly two games.

You have a $10\times 10$ grid of squares. You write a number in each square as follows: you write $1$, $2$, $3$ ,$ ...$ ,$10$ from left to right across the top row, then $11$, $12$, $...$, $20$ across the second row, and so on, ending with a $100$ in the bottom right square. You then write a second number in each square, writing $1$, $2$, $3$ ,$ ...$ ,$10$ in the first column (from top to bottom), then $11$, $12$, $...$, $20$ in the second column, and so forth. When this process is finished, how many squares will have the property that their two numbers sum to $101$?

There are $26$ cards and each one has a number written on it. There are two with $1$, two with $2$, two with $3$, and so on up to two with $12$ and two with $13$. You have to distribute the $26$ cards in piles so that the following two conditions are met: $\bullet$ If two cards have the same number they are in the same pile. $\bullet$ No pile contains a card whose number is equal to the sum of the numbers of two cards in that same pile. Determine what is the minimum number of stacks to make. Give an example with the distribution of the cards for that number of stacks and justify why it is impossible to have fewer stacks. Clarification: Two squares are [i]neighbors [/i] if they have a common side.

Anna and Bob are playing the following game: The number $2$ is initially written on the blackboard. With Anna playing first, they alternately double the number currently written on the blackboard or square it. The person who first writes on the blackboard a number greater than $2023^{10}$ is the winner. Determine which player has a winning strategy.

$100$ ones are written in a circle. Petya and Vasya take turns making \( 10^{10} \) moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by $2$. Vasya chooses $10$ consecutive numbers and increases each by $1$. They alternate turns, starting with Petya. Prove that Vasya can act in such a way that after each of his moves, there are always at least five positive numbers, regardless of how Petya plays. \\

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

On a competition called [i]"Mathematical duels"[/i] students were given $n$ problems and each student solved exactly 3 of them. For each two students there is at most one problem that is solved from both of them. Prove that, if $s\in \mathbb{N}$ is a number for which $s^2-s+1<2n$, then there are $s$ problems among the $n$, no three of which solved by one student.

Suppose we have $2016$ points in a $2$-dimensional plane such that no three lie on a line. Two quadrilaterals are not disjoint if they share an edge or vertex, or if their edges intersect. Show that there are at least $504$ quadrilaterals with vertices among these points such that any two of the quadrilaterals are disjoint.

A square $ n \times n $, ($ n> 2 $) contains nonzero real numbers. It is known that every number is exactly $ k $ times smaller than the sum of all the numbers in its row or sum of all number in its column. For which real numbers $ k $ is this possible?

As shown in the following figure, there is a line segment consisting of five line segments $AB, BC, CD, DE, and EA$ and $10$ intersection points of these five line segments. Find the number of ways to write $1$ or $2$ at each of the $10$ vertices so that the following conditions are satisfied. $\bigstar$ The sum of the four numbers written on each line segment $AB, BC, CD, DE, and EA$ is the same.

You place $n^2$ indistinguishable pieces on an $n\times n$ chessboard, where $n=2^{90}\approx 1.27\times10^{27}$. Of these pieces, $n$ of them are slightly lighter than usual, while the rest are all the same standard weight, but you are unable to discern this simply by feeling the pieces.\\ However, beneath each row and column of the chessboard, you have set up a scale, which, when turned on, will tell you [i]only[/i] whether the average weight of all the pieces on that row or column is the standard weight, or lighter than standard. On a given step, you are allowed to rearrange every piece on the chessboard, and then turn on all the scales simultaneously, and record their outputs, before turning them all off again. (Note that you can only turn on the scales if all $n^2$ pieces are placed in different squares on the board.) Find an algorithm that, in at most $k$ steps, will always allow you to rearrange the pieces in such a way that every row and column measures lighter than standard on the final step. An algorithm that completes in at most $k$ steps will be awarded: 1 pt for $k>10^{55}$ 10 pts for $k=10^{55}$ 30 pts for $k=10^{30}$ 50 pts for $k=10^{28}$ 65 pts for $k=10^{20}$ 80 pts for $k=10^5$ 90 pts for $k=2021$ 100 pts for $k=500$

Ashwin the frog is traveling on the $xy$-plane in a series of $2^{2017} -1$ steps, starting at the origin. At the $n^{th}$ step, if $n$ is odd, then Ashwin jumps one unit to the right. If $n$ is even, then Ashwin jumps $m$ units up, where $m$ is the greatest integer such that $2^m$ divides $n$. If Ashwin begins at the origin, what is the area of the polygon bounded by Ashwin’s path, the line $x = 2^{2016}$, and the $x$-axis?

A Knight always tells the truth. A Knave always lies. A Normal may either lie or tell the truth. You are allowed to ask questions that can be answered with ''yes" or ''no", such as ''is this person a Normal?" (a) There are three people in front of you. One is a Knight, another one is a Knave, and the third one is a Normal. They all know the identities of one another. How can you too learn the identity of each? (1) (b) There are four people in front of you. One is a Knight, another one is a Knave, and the other two are Normals. They all know the identities of one another. Prove that the Normals may agree in advance to answer your questions in such a way that you will not be able to learn the identity of any of the four people. (3)

In the Flower-city there are $ n$ squares and $ m$ streets, where $ m \geq n \plus{} 1$. Each street connects two squares and does not pass through other squares. According to a tradition in the city, each street is named either blue or red. Every year, a square is selected and the names of all streets emanating from that square are changed. Show that the streets can be initially named in such a way that, no matter how the names will be changed, the streets will never all have the same name.

Let $A_1 A_2...A_n$ be a convex $n$-gon. What’s the number of $m$-gons with vertices from $A_1,A_2,...,A_n$ such that between each two adjacent vertices of the $m$-gon there are at least $k$ vertices from the $n$-gon?

At the start of this problem, six frogs are sitting at each of the six vertices of a regular hexagon, one frog per vertex. Every minute, we choose a frog to jump over another frog using one of the two rules illustrated below. If a frog at point $F$ jumps over a frog at point $P$ the frog will land at point $F'$ such that $F, P,$ and $F'$ are collinear and: - using Rule 1, $F'P = 2FP$. - using Rule 2, $F'P = FP/2$. [img]https://cdn.artofproblemsolving.com/attachments/7/0/2936bda61eb60c7b89bcd579386041022ba81f.png[/img] It is up to us to choose which frog to take the leap and which frog to jump over. (a) If we only use Rule 1, is it possible for some frog to land at the center of the original hexagon after a finite amount of time? (b) If both Rule 1 and Rule 2 are allowed (freely choosing which rule to use, which frog to jump, and which frog it jumps over), is it possible for some frog to land at the center of the original hexagon after a finite amount of time?

Let's call a checkered polygon a [i]strip[/i], which can be traversed entirely, starting from some of its cells and then moving only in two directions - up or to the right. Several such strips can be inserted into each other by shifting by a vector $(-1.1)$. Prove that for any strip consisting of an even number of cells, there is such an odd $k$ that if you combine $k$ of the same strips by inserting them sequentially into each other, then the resulting polygon can be divided along the grid lines into two equal parts. [i]Proposed by I. Markelov, S. Markelov[/i]

There are several scientists collaborating in Niichavo. During an $ 8$-hour working day, the scientists went to cafeteria, possibly several times.It is known that for every two scientist, the total time in which exactly one of them was in cafeteria is at least $ x$ hours ($ x>4$). What is the largest possible number of scientist that could work in Niichavo that day,in terms of $ x$?

There are five smart kids sitting around a round table. Their teacher says: "I gave a few apples to some of you, and none of you have the same amount of apple. Also each of you will know the amount of apple that the person to your left and the person to your right has." The teacher tells the total amount of apples, then asks the kids to guess the difference of the amount of apple that the two kids in front of them have. $a)$ If the total amount of apples is less than $16$, prove that at least one of the kids will guess the difference correctly. $b)$ Prove that the teacher can give the total of $16$ apples such that no one can guess the difference correctly.

On a soccer tournament with $n\ge 3$ teams taking part, several matches are played in such a way that among any three teams, some two play a match. $(a)$ If $n=7$, find the smallest number of matches that must be played. $(b)$ Find the smallest number of matches in terms of $n$.

Three piles of stones are given, initially containing 2000, 4000, and 4899 stones respectively. Ali and Baba play the following game, taking turns, with Ali starting first. In one move, a player can choose two piles and transfer some stones from one pile to the other, provided that at the end of the move, the pile from which the stones are moved has no fewer stones than the pile to which the stones are moved. The player who cannot make a move loses. Does either player have a winning strategy, and if so, who?

Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$. A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$. A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$. Now put a pawn on $(0, 0)$. You can make a (nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B. Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.

On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as [i]appropriate[/i] while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called [i]perspective[/i] if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called [i]universal[/i] if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.