Is it possible to partition the set of positive integer numbers into two classes, none of which contains an infinite arithmetic sequence (with a positive ratio)? What is we impose the extra condition that in each class $\mathcal{C}$ of the partition, the set of difference \begin{align*} \left\{ \min \{ n \in \mathcal{C} \mid n >m \} -m \mid m \in \mathcal{C} \right \} \end{align*} be bounded?

Let $n \ge 3$ be an integer. A [i]circle dance[/i] is a dance that is performed according to the following rule: On the floor, $n$ points are marked at equal distances along a large circle. At each of these points is a sheet of paper with an arrow pointing either clockwise or counterclockwise. One of the points is labeled "Start". The dancer starts at this point. In each step, he first changes the direction of the arrow at his current position and then moves to the next point in the new direction of the arrow. a) Show that each circle dance visits each point infinitely often. b) How many different circle dances are there? Two circle dances are considered to be the same if they differ only by a finite number of steps at the beginning and then always visit the same points in the same order. (The common sequence of steps may begin at different times in the two dances.) [i](Birgit Vera Schmidt)[/i]

Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) ...(1 - 2008x)$.

Let $Q=\{0,1\}^n$, and let $A$ be a subset of $Q$ with $2^{n-1}$ elements. Prove that there are at least $2^{n-1}$ pairs $(a,b)\in A\times (Q\setminus A)$ for which sequences $a$ and $b$ differ in only one term.

In plane there are $n$ noncollinear points $A_1$, $A_2$,...,$A_n$. Prove that there exist a line which passes through exactly two of these points

Let it \(k\) be a fixed positive integer. Alberto and Beralto play the following game: Given an initial number \(N_0\) and starting with Alberto, they alternately do the following operation: change the number \(n\) for a number \(m\) such that \(m < n\) and \(m\) and \(n\) differ, in its base-2 representation, in exactly \(l\) consecutive digits for some \(l\) such that \(1 \leq l \leq k\). If someone can't play, he loses. We say a non-negative integer \(t\) is a [i]winner[/i] number when the gamer who receives the number \(t\) has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player. Else, we call it [i]loser[/i]. Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\)

There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

We say that a graph $G$ is [i]divisive[/i], if we can write a positive integer on each of its vertices such that all the integers are distinct, and any two of these integers divide each other if and only if there is an edge running between them in $G$. Which Platonic solids form a divisive graph? [img]https://cdn.artofproblemsolving.com/attachments/1/5/7c81439ee148ccda09c429556e0740865723e0.png[/img]

An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.

For an integer $n\geq 3$, place $n$ points on the plane in such a way that all the distances between the points are at most one and exactly $n$ of the pairs of points have the distance one.

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.

Nestor ordered Juan to do the following work: draw a circle, draw one of its diameters and mark the extreme points of the diameter with the numbers 1 and 2 respectively. Place 100 points in each of the semicircles that determines the diameter layout (different from the ends of the diameter) and mark these points randomly with the numbers $1$ and $2$. To finish, paint red all small segments that have different markings on their extremes. After a certain amount of time passed, Juan finished the work and told Nestor that “he painted 47 segments red.” Prove that if Juan made no mistakes, what he said is false.

[b]p1.[/b] Let $D(n)$ denote the number of positive factors of the integer $n$. For example, $D(6) = 4$ , since the factors of $6$ are $1, 2, 3$ , and $6$ . Note that $D(n) = 2$ if and only if $n$ is a prime number. (a) Describe the set of all solutions to the equation $D(n) = 5$ . (b) Describe the set of all solutions to the equation $D(n) = 6$ . (c) Find the smallest $n$ such that $D(n) = 21$ . [b]p2.[/b] At a party with $n$ married couples present (and no one else), various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening Mr. Jones asked everyone else, including his wife, how many hands he or she had shaken. To his surprise, he got a different answer from each person. Determine the number of hands that Mr. Jones shook that evening, (a) if $n = 2$ . (b) if $n = 3$ . (c) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [b]p3.[/b] Let $n$ be a positive integer. A square is divided into triangles in the following way. A line is drawn from one corner of the square to each of $n$ points along each of the opposite two sides, forming $2n + 2$ nonoverlapping triangles, one of which has a vertex at the opposite corner and the other $2n + 1$ of which have a vertex at the original corner. The figure shows the situation for $n = 2$ . Assume that each of the $2n + 1$ triangles with a vertex in the original corner has area $1$. Determine the area of the square, (a) if $n = 1$ . (b) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [img]https://cdn.artofproblemsolving.com/attachments/1/1/62a54011163cc76cc8d74c73ac9f74420e1b37.png[/img] [b]p4.[/b] Arthur and Betty play a game with the following rules. Initially there are one or more piles of stones, each pile containing one or more stones. A legal move consists either of removing one or more stones from one of the piles, or, if there are at least two piles, combining two piles into one (but not removing any stones). Arthur goes first, and play alternates until a player cannot make a legal move; the player who cannot move loses. (a) Determine who will win the game if initially there are two piles, each with one stone, assuming that both players play optimally. (b) Determine who will win the game if initially there are two piles, each with $n$ stones, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . (c) Determine who will win the game if initially there are $n$ piles, each with one stone, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . [b]p5.[/b] Suppose $x$ and $y$ are real numbers such that $0 < x < y$. Define a sequence$ A_0 , A_1 , A_2, A_3, ...$ by-setting $A_0 = x$ , $A_1 = y$ , and then $A_n= |A_{n-1}| - A_{n-2}$ for each $n \ge 2$ (recall that $|A_{n-1}|$ means the absolute value of $A_{n-1}$ ). (a) Find all possible values for $A_6$ in terms of $x$ and $y$ . (b) Find values of $x$ and $y$ so that $A_{1987} = 1987$ and $A_{1988} = -1988$ (simultaneously). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We have $k$ switches arranged in a row, and each switch points up, down, left, or right. Whenever three successive switches all point in different directions, all three may be simultaneously turned so as to point in the fourth direction. Prove that this operation cannot be repeated infinitely many times.

Let $A$ be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of $A$ satisfying $x < y$ and $x + y = z$.

Prior to the game John selects an integer greater than $100$. Then Mary calls out an integer $d$ greater than $1$. If John's integer is divisible by $d$, then Mary wins. Otherwise, John subtracts $d$ from his number and the game continues (with the new number). Mary is not allowed to call out any number twice. When John's number becomes negative, Mary loses. Does Mary have a winning strategy?

Let $P =\{(a, b) / a, b \in \{1, 2, ..., n\}, n \in N\}$ be a set of point of the Cartesian plane and draw horizontal, vertical, or diagonal segments, of length $1$ or $\sqrt 2$, so that both ends of the segment are in $P$ and do not intersect each other. Furthermore, for each point $(a, b)$ it is true that i) if $a + b$ is a multiple of $3$, then it is an endpoint of exactly $3$ segments. ii) if $a + b$ is an even not multiple of $3$, then it is an endpoint of exactly $2$ segments. iii) if $a + b$ is an odd not multiple of $3$, then it is endpoint of exactly $1$ segment. a) Check that with $n = 6$ it is possible to satisfy all the conditions. b) Show that with $n = 2015$ it is not possible to satisfy all the conditions.

Let $n$ be a positive integer. There is a board of $(n + 1) \times (n + 1)$ whose squares are numbered in a diagonal pattern, as as the picture shows. Chepito starts from the lower left square, and moving only up or to the right until he reaches the upper right box. During his tour, Chepito writes down the number of each box on the which made a change of direction, and in the end calculates the sum of all the numbers entered. Determine the maximum value of this sum. [img]https://cdn.artofproblemsolving.com/attachments/e/d/f9dc43092a1407d6fe6f1b2c741af015079946.png[/img]

From all possible permutations from $(a_1,a_2,...,a_n)$ from the set $\{1,2,..,n\}$, $n\geq 1$, consider the sets that satisfies the $2(a_1+a_2+...+a_m)$ is divisible by $m$, for every $m=1,2,...,n$. Find the total number of permutations.

We say that a positive integer $n$ is $m$[i]-expressible[/i] if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$, exponentiation $^\wedge$, and concatenation $\oplus$. For example, $5625$ is $3$-expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$. Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$-expressible? [i]Proposed by Krit Boonsiriseth[/i]

The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements?

Let $n \ge 2$ be a positive integer. Each square of an $n\times n$ board is coloured red or blue. We put dominoes on the board, each covering two squares of the board. A domino is called [i]even [/i] if it lies on two red or two blue squares and [i]colourful [/i] if it lies on a red and a blue square. Find the largest positive integer $k$ having the following property: regardless of how the red/blue-colouring of the board is done, it is always possible to put $k$ non-overlapping dominoes on the board that are either all [i]even [/i] or all [i]colourful[/i].

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

Given an $n \times n$ ($n \ge 3$) square table with one of the following unit vectors $\uparrow, \downarrow, \leftarrow, \rightarrow$ in any its cell (the vectors are parallel to the sides and the middles of them coincide with the centers of the cells). Per move a beetle creeps from one cell to another in accordance with the vector’s direction. If the beetle starts from any cell, then it comes back to this cell after some number of moves. The vectors are directed so that they do not allow the beetle to leave the table. Is it possible that the sum of all vectors at any row (except for the first one and the last one) is equal to the vector that is parallel to this row, and the sum of all vectors at any column (except for the first one and the last one) is equal to the vector that is parallel to this column ? (D. Dudko)

There are $12$ points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?