Show that any graph with $6$ points has a triangle or three points which are not joined to each other.

A rook is randomly placed on an otherwise empty $8 \times 8$ chessboard. Owen makes moves with the rook by randomly choosing $1$ of the $14$ possible moves. Find the expected value of the number of moves it takes Owen to move the rook to the top left square. Note that a rook can move any number of squares either in the horizontal or vertical direction each move.

For a given positive integer $n$, we consider the set $M$ of all intervals of the form $[l, r]$, where the integers $l$ and $r$ satisfy the condition $0 \leq l < r \leq n$. What largest number of elements of $M$ can be chosen so that each chosen interval completely contains at most one other selected interval? [i]Proposed by Anton Trygub[/i]

There is a frog jumping on a $ 2k \times 2k$ chessboard, composed of unit squares. The frog's jumps are $ \sqrt{1 \plus{} k^2}$ long and they carry the frog from the center of a square to the center of another square. Some $ m$ squares of the board are marked with an $ \times$, and all the squares into which the frog can jump from an $ \times$'d square (whether they carry an $ \times$ or not) are marked with an $ \circ$. There are $ n$ $ \circ$'d squares. Prove that $ n \ge m$.

Given parallelogram $OABC$ in the coodinate with $O$ the origin and $A,B,C$ be lattice points. Prove that for all lattice point $P$ in the internal or boundary of $\triangle ABC$, there exists lattice points $Q,R$(can be the same) in the internal or boundary of $\triangle OAC$ with $\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}$.

On a board of $8 \times 8$ exists $64$ kings, all initially placed in different squares. Alnardo and Bernaldo play alternately, with Arnaldo starting. On each move, one of the two players chooses a king and can move it one square to the right, one square up, or one square up to the right. In the event that a king is moved to an occupied square, both kings are removed from the game. The player who can remove two of the last kings or leave one last king in the upper right corner wins the game. Which of the two players can ensure victory?

Write down some numbers $a_1,a_2,\ldots, a_n$ from left to right on a line. Step 1, we write $a_1+a_2$ between $a_1,a_2$; $a_2+a_3$ between $a_2,a_3$, …, $a_{n-1}+a_n$ between $a_{n-1},a_n$, and then we have new sequence $b=(a_1, a_1+a_2,a_2,a_2+a_3,a_3, \ldots, a_{n-1}, a_{n-1}+a_n, a_n)$. Step 2, we do the same thing with sequence b to have the new sequence c again…. And so on. If we do 2013 steps, count the number of the number 2013 appear on the line if a) $n=2$, $a_1=1, a_2=1000$ b) $n=1000$, $a_i=i, i=1,2\ldots, 1000$ Sorry for my bad English [color=#008000]Moderator says: alternate phrasing here: https://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=516134[/color]

[b]p1.[/b] Compute $45\times 45 - 6$. [b]p2.[/b] Consecutive integers have nice properties. For example, $3$, $4$, $5$ are three consecutive integers, and $8$, $9$, $10$ are three consecutive integers also. If the sum of three consecutive integers is $24$, what is the smallest of the three numbers? [b]p3.[/b] How many positive integers less than $25$ are either multiples of $2$ or multiples of $3$? [b]p4.[/b] Charlotte has $5$ positive integers. Charlotte tells you that the mean, median, and unique mode of his five numbers are all equal to $10$. What is the largest possible value of the one of Charlotte's numbers? [b]p5.[/b] Mr. Meeseeks starts with a single coin. Every day, Mr. Meeseeks goes to a magical coin converter where he can either exchange $1$ coin for $5$ coins or exchange $5$ coins for $3$ coins. What is the least number of days Mr. Meeseeks needs to end with $15$ coins? [b]p6.[/b] Twelve years ago, Violet's age was twice her sister Holo's age. In $7$ years, Holo's age will be $13$ more than a third of Violet's age. $3$ years ago, Violet and Holo's cousin Rindo's age was the sum of their ages. How old is Rindo? [b]p7.[/b] In a $2 \times 3$ rectangle composed of $6$ unit squares, let $S$ be the set of all points $P$ in the rectangle such that a unit circle centered at $P$ covers some point in exactly $3$ of the unit squares. Find the area of the region $S$. For example, the diagram below shows a valid unit circle in a $2 \times 3$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/d/9/b6e00306886249898c2bdb13f5206ced37d345.png[/img] [b]p8.[/b] What are the last four digits of $2^{1000}$? [b]p9.[/b] There is a point $X$ in the center of a $2 \times 2 \times 2$ box. Find the volume of the region of points that are closer to $X$ than to any of the vertices of the box. [b]p10.[/b] Evaluate $\sqrt{37 \cdot 41 \cdot 113 \cdot 290 - 4319^2}$ [b]p11.[/b] (Estimation) A number is abundant if the sum of all its divisors is greater than twice the number. One such number is $12$, because $1+2+3+4+6+12 = 28 > 24$: How many abundant positive integers less than $20190$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there exist infinitely many pairwisely disjoint sets $A(1), A(2),...,A(2014)$ which are not empty, whose union is the set of positive integers and which satisfy the following condition: For arbitrary positive integers $a$ and $b$, at least two of the numbers $a$, $b$ and $GCD(a,b)$ belong to one of the sets $A(1), A(2),...,A(2014)$.

An Indian raga has two kinds of notes: a short note, which lasts for $1$ beat and a long note, which lasts for $2$ beats. For example, there are $3$ ragas which are $3$ beats long; $3$ short notes, a short note followed by a long note, and a long note followed by a short note. How many Indian ragas are 11 beats long?

Alice and Bob are playing a game on a plane consisting of $72$ cells arranged in circle. At the beginning of the game, Bob places a stone on some of the cells. Then, in every round first Alice picks one empty cell and then Bob must move a stone from one of the two neighboring cells on this cell. If he is unable to do that, game ends. Determine the smallest number of stones he has to place in the beginning so he has a strategy to make the game last for at least $2023$ rounds.

A complete graph is in a plane such that no three of its vertices are collinear. The edges of the graph, which are straight segments connecting the vertices, are colored with two colors. Prove that there is a non-self-intersecting spanning tree consisting of edges of the same color.

In lawn-tennis the player who scores at least four points, while his opponent scores at least two points less, wins a game. The player who wins at least six games, while his opponent wins at least two games less, wins a set. What minimum percentage of all points does the winner have to score in a set?

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Jerry places at most one rook in each cell of a $2025 \times 2025$ grid of cells. A rook [i]attacks[/i] another rook if the two rooks are in the same row or column and there are no other rooks between them. Determine, with proof, the maximum number of rooks Jerry can place on the grid such that no rook attacks $4$ other rooks.

The $52$ cards of a standard deck are placed in a $13\times 4$ array. If every two adjacent cards, vertically or horizontally, have the same suit or have the same value, prove that all $13$ cards of the same suit are in the same row.

There are 16 boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a 1$0$ day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)

A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\times 8 \times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizontal and vertical plates so that each box has at least one face which is not touching other boxes. What is the least number of plates needed for this purpose?

In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.

Jeff rolls a standard $6$ sided die repeatedly until he rolls either all of the prime numbers possible at least once, or all the of even numbers possible at least once. Find the probability that his last roll is a $2$.

An equilateral triangle of side $n$ has been divided into little equilateral triangles of side $1$ in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the $\frac{(n+1)(n+2)}{2}$ vertices. What is the minimum number of times the path can change its direction? The figure below shows a valid path on a triangular board of side $4$, with exactly $9$ changes of direction. [asy] unitsize(30); pair h = (1, 0); pair v = dir(60); pair d = dir(120); for(int i = 0; i < 4; ++i) { draw(i*v -- i*v + (4 - i)*h); draw(i*h -- i*h + (4 - i)*v); draw((i + 1)*h -- (i + 1)*h + (i + 1)*d); } draw(h + v -- v -- (0, 0) -- 2*h -- 2*h + v -- h + 2*v -- 2*v -- 4*v -- 3*h + v -- 3*h -- 4*h, linewidth(2)); draw(3*h -- 4*h, EndArrow); fill(circle(h + v, 0.1)); [/asy] [i]Proposed by Oriol Solé[/i]

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

In every cell of an $n\times n$ board is written $1$ or $-1$. At each step we may choose any of the $4n-2$ diagonals of the board and change the signs of all the numbers on that diagonal. Determine the number of initial configurations from which, after a finite number of steps, we may arrive at a configuration where all products of numbers on rows and columns equal to $1$. [i]Proposed by Pavel Ciurea[/i]

Let $n, k$ be natural numbers, $1 \leq k < n$. In each vertex of a regular polygon with $n$ sides is written $1$ or $-1$. At each step we choose $k$ consecutive vertices and change their signs. Is it possible that, starting from a certain configuration and by doing the operation a few times to obtain any other configuration?

Let $m> 3$ be an integer. At a camp there are more than $m$ participants. The camp manager discovers that every time he picks out the camp participants, they say they have exactly one mutual friend among the participants. Which is the largest possible number of participants at the camp? (If $A$ is a friend of $B, B$ is also a friend of $A$. A person is not considered a friend of himself.)