Is there a colouring of all positive integers in three colours so that for each positive integer the numbers of its divisors of any two colours differ at most by $2?$

Marisa has a collection of $2^8-1=255$ distinct nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process $2^8-2=254$ times until there is only one set left in the collection. What is the expected size of this set?

Five numbers are chosen from $\{1, 2, \ldots, n\}$. Determine the largest $n$ such that we can always pick some of the 5 chosen numbers so that they can be made into two groups whose numbers have the same sum (a group may contain only one number).

Let $f(n)$ take in a nonnegative integer $n$ and return an integer between $0$ and $n-1$ at random (with the exception being $f(0)=0$ always). What is the expected value of $f(f(22))$?

There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.

Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

Consider lattice points of a $6*7$ grid.We start with two points $A,B$.We say two points $X,Y$ connected if one can reflect several times WRT points $A,B$ and reach from $X$ to $Y$.Over all choices of $A,B$ what is the minimum number of connected components?

[b]p1.[/b] We define $a \oplus b = \frac{ab}{a+b}$. Compute $(3 \oplus 5) \oplus (5 \oplus 4)$. [b]p2.[/b] Let $ABCD$ be a quadrilateral with $\angle A = 45^o$ and $\angle B = 45^o$. If $BC = 5\sqrt2$, $AD = 6\sqrt2$, and $AB = 18$, find the length of side $CD$. [b]p3.[/b] A positive real number $x$ satisfies the equation $x^2 + x + 1 + \frac{1}{x }+\frac{1}{x^2} = 10$. Find the sum of all possible values of $x + 1 + \frac{1}{x}$. [b]p4.[/b] David writes $6$ positive integers on the board (not necessarily distinct) from least to greatest. The mean of the first three numbers is $3$, the median of the first four numbers is $4$, the unique mode of the first five numbers is $5$, and the range of all 6 numbers is $6$. Find the maximum possible value of the product of David’s $6$ integers. [b]p5.[/b] Let $ABCD$ be a convex quadrilateral such that $\angle A = \angle B = 120^o$ and $\angle C = \angle D = 60^o$. There exists a circle with center $I$ which is tangent to all four sides of $ABCD$. If $IA \cdot IB \cdot IC \cdot ID = 240$, find the area of quadrilateral $ABCD$. [b]p6.[/b] The letters $EXETERMATH$ are placed into cells on an annulus as shown below. How many ways are there to color each cell of the annulus with red, blue, green, or yellow such that each letter is always colored the same color and adjacent cells are always colored differently? [img]https://cdn.artofproblemsolving.com/attachments/3/5/b470a771a5279a7746c06996f2bb5487c33ecc.png[/img] [b]p7.[/b] Let $ABCD$ be a square, and let $\omega$ be a quarter circle centered at $A$ passing through points $B$ and $D$. Points $E$ and $F$ lie on sides $BC$ and $CD$ respectively. Line $EF$ intersects $\omega$ at two points, $G$ and $H$. Given that $EG = 2$, $GH = 16$ and $HF = 9$, find the length of side $AB$. [b]p8.[/b] Let x be equal to $\frac{2022! + 2021!}{2020! + 2019! + 2018!}$ . Find the closest integer to $2\sqrt{x}$. [b]p9.[/b] For how many ordered pairs of positive integers $(m, n)$ is the absolute difference between $lcm(m, n)$ and $gcd(m, n)$ equal to $2023$? [b]p10.[/b] There are $2023$ distinguishable frogs sitting on a number line with one frog sitting on $i$ for all integers $i$ between $-1011$ and $1011$, inclusive. Each minute, every frog randomly jumps either one unit left or one unit right with equal probability. After $1011$ minutes, over all possible arrangements of the frogs, what is the average number of frogs sitting on the number $0$? [b]p11.[/b] Albert has a calculator initially displaying $0$ with two buttons: the first button increases the number on the display by one, and the second button returns the square root of the number on the display. Each second, he presses one of the two buttons at random with equal probability. What is the probability that Albert’s calculator will display the number $6$ at some point? [b]p12.[/b] For a positive integer $k \ge 2$, let $f(k)$ be the number of positive integers $n$ such that n divides $(n-1)!+k$. Find $$f(2) + f(3) + f(4) + f(5) + ... + f(100).$$ [b]p13.[/b] Mr. Atf has nine towers shaped like rectangular prisms. Each tower has a $1$ by $1$ base. The first tower as height $1$, the next has height $2$, up until the ninth tower, which has height $9$. Mr. Atf randomly arranges these $9$ towers on his table so that their square bases form a $3$ by $3$ square on the surface of his table. Over all possible solids Mr. Atf could make, what is the average surface area of the solid? [b]p14.[/b] Let $ABCD$ be a cyclic quadrilateral whose diagonals are perpendicular. Let $E$ be the intersection of $AC$ and $BD$, and let the feet of the altitudes from $E$ to the sides $AB$, $BC$, $CD$, $DA$ be $W, X, Y , Z$ respectively. Given that $EW = 2EY$ and $EW \cdot EX \cdot EY \cdot EZ = 36$, find the minimum possible value of $\frac{1}{[EAB]} +\frac{1}{[EBC]}+\frac{1}{[ECD]} +\frac{1}{[EDA]}$. The notation $[XY Z]$ denotes the area of triangle $XY Z$. [b]p15.[/b] Given that $x^2 - xy + y^2 = (x + y)^3$, $y^2 - yz + z^2 = (y + z)^3$, and $z^2 - zx + x^2 = (z + x)^3$ for complex numbers $x, y, z$, find the product of all distinct possible nonzero values of $x + y + z$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Calculate \[T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!}\] where the sum is taken over all 1994-tuples of the numbers $n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\}$ satisfying $n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots + 1994 \cdot n_{1994} = 1994.$

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

Albert and Kevin are playing a game. Kevin has a $10\%$ chance of winning any given round in the match. If Kevin wins the first game, he wins the match. If not, he requests that the match be extended to a best of $3$. If he wins the best of $3$, he wins the match. If not, then he requests the match be extended to a best of $5$, and so forth. What is the probability that Kevin eventually wins the match? (A best of $2n+ 1$ match consists of a series of rounds. The first person to reach $n + 1$ winning games wins the match)

Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$, determine maximum of $A$

For a natural number $n \ge 3$, we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$-gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$?

Let $n > 1$ be an integer. In a [i]configuration[/i] of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell [i]good[/i] if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations. [i]Proposed by Melek Güngör, Turkey[/i]

Let $n$, $d$ be positive integers such that $d>\frac{n}{2}$. Suppose $a_1, a_2,\cdots,a_{d+2}$ is a sequence of integers satisfying $a_{d+1}=a_1$, $a_{d+2}=a_2$, and for all indices $1\le i_1<i_2<\cdots <i_s\le d$, $$a_{i_1}+a_{i_2}+\cdots+a_{i_s}\not\equiv 0\pmod n$$ Prove that there exists $1\le i\le d$ such that $$a_{i+1}\equiv a_i \pmod n \quad \text{or} \quad a_{i+1}\equiv a_i+a_{i+2} \pmod n$$ [i]Proposed by Yeoh Zi Song[/i]

[b]p1.[/b] $17$ rooks are placed on an $8\times 8$ chess board. Prove that there must be at least one rook that is attacking at least $2$ other rooks. [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins? [b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h, i$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc} & & a & b & c \\ x & & & d & e \\ \hline & f & a & c & c \\ + & g & h & i & \\ \hline f & f & f & c & c \\ \end{tabular}$ [b]p4.[/b] Pinocchio rode a bicycle for $3.5$ hours. During every $1$-hour period he went exactly $5$ km. Is it true that his average speed for the trip was $5$ km/h? Explain your reasoning. [b]p5.[/b] Let $a, b, c$ be odd integers. Prove that the equation $ax^2 + bx + c = 0$ cannot have a rational solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three of the $16$ squares from a $4 \times 4$ grid of squares are selected at random. The probability that at least one corner square of the grid is selected is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $ \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular} $

$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

The numbers $1, 2, ..., 2020$ and $2021$ are written on a blackboard. The following operation is executed: Two numbers are chosen, both are erased and replaced by the absolute value of their difference. This operation is repeated until there is only one number left on the blackboard. (a) Show that $2021$ can be the final number on the blackboard. (b) Show that $2020$ cannot be the final number on the blackboard. (Karl Czakler)

For a positive integer $n,$ let $P_n$ be the set of sequences of $2n$ elements, each $0$ or $1,$ where there are exactly $n$ $1$’s and $n$ $0$’s. I choose a sequence uniformly at random from $P_n.$ Then, I partition this sequence into maximal blocks of consecutive $0$’s and $1$’s. Define $f(n)$ to be the expected value of the sum of squares of the block lengths of this uniformly random sequence. What is the largest integer value that $f(n)$ can take on?

Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins): At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture. Is there a winning strategy available for Sasha?

Initially, a natural number $n$ is written on the blackboard. Then, at each minute, [i]Neymar[/i] chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that [i]Neymar[/i] will never be able to write on the blackboard?

Suppose that $n$ squares of an infinite square grid are colored grey, and the rest are colored white. At each step, a new grid of squares is obtained based on the previous one, as follows. For each location in the grid, examine that square, the square immediately above, and the square immediately to the right. If there are two or three grey squares among these three, then in the next grid, color that location grey, otherwise, color it white. Prove that after at most n steps all the squares in the grid will be white. Below is an example with $n = 4$. The first grid shows the initial configuration, and the second grid shows the configuration after one step. [img]https://cdn.artofproblemsolving.com/attachments/1/a/87f7e3892cdb45fb3529127234aae2cea08749.png[/img]

How many divisors of $30^{2003}$ are there which do not divide $20^{2000}$?