There are $3n$ participants in the Mathematical Olympiad competition. They are assigned seats in three rows, with $n$ seats in each, and are admitted into the hall one at a time, after which they immediately take their seats. Calculate the probability that until the last competitor takes his seat, at any moment for each two rows the difference in the number of players sitting in them is no greater than 1.

Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression \[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\]

Let $m$ be a nonnegative integer. Show that the number of tilings of a $(2m + 2) \times (2m + 2)$ grid of squares by $1 \times 2$ or $2 \times 1$ rectangles is at least $$2 \cdot 2^{\frac{5}{2}m} \cdot 5120^{\frac{1}{8}m^2}.$$ [i]Proposed by Milan Haiman[/i]

In a checked $ 17\times 17$ table, $ n$ squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least $ 6$ of the squares in some line are black, then one can paint all the squares of this line in black. Find the minimal value of $ n$ such that for some initial arrangement of $ n$ black squares one can paint all squares of the table in black in some steps.

Let $S$ be the set of lattice points $(x,y) \in \mathbb{Z}^2$ such that $-10\leq x,y \leq 10$. Let the point $(0,0)$ be $O$. Let Scotty the Dog's position be point $P$, where initially $P=(0,1)$. At every second, consider all pairs of points $C,D \in S$ such that neither $C$ nor $D$ lies on line $OP$, and the area of quadrilateral $OCPD$ (with the points going clockwise in that order) is $1$. Scotty finds the pair $C,D$ maximizing the sum of the $y$ coordinates of $C$ and $D$, and randomly jumps to one of them, setting that as the new point $P$. After $50$ such moves, Scotty ends up at point $(1, 1)$. Find the probability that he never returned to the point $(0,1)$ during these $50$ moves. [i]Proposed by David Tang[/i]

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$.

Let $n\geqslant3$ be an integer. Let $A_1A_2\ldots A_n$ be a convex $n$-gon. Consider a line $g$ through $A_1$ that does not contain a further vertice of the $n$-gon. Let $h$ be the perpendicular to $g$ through $A_1$. Project the $n$-gon orthogonally on $h$. For $j=1,\ldots,n$, let $B_j$ be the image of $A_j$ under this projection. The line $g$ is called admissible if the points $B_j$ are pairwise distinct. Consider all convex $n$-gons and all admissible lines $g$. How many different orders of the points $B_1,\ldots,B_n$ are possible?

Given an ordered $n$-tuple $A=(a_1,a_2,\cdots ,a_n)$ of real numbers, where $n\ge 2$, we define $b_k=\max{a_1,\ldots a_k}$ for each k. We define $B=(b_1,b_2,\cdots ,b_n)$ to be the “[i]innovated tuple[/i]” of $A$. The number of distinct elements in $B$ is called the “[i]innovated degree[/i]” of $A$. Consider all permutations of $1,2,\ldots ,n$ as an ordered $n$-tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to $2$

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

Let ${n \choose k}$ be the number of ways that $k$ objects can be chosen (regardless of order) from a set of $n$ objects. Prove that if positive integers k and l are greater than $1$ and less than $n$, then integers ${n \choose k}$ and ${n \choose l}$ have a common divisor greater than $1$.

Some of the $80960$ lattice points in a $40\times2024$ lattice are coloured red. It is known that no four red lattice points are vertices of a rectangle with sides parallel to the axes of the lattice. What is the maximum possible number of red points in the lattice?

The computer "Intel stump-V" can do only one operation with a number: add 1 to it, then rearrange all the zeros in the decimal representation to the end and rearrenge the left digits in any order. (For example from 1004 you could get 1500 or 5100). The number $12345$ was written on the computer and after performing 400 operations, the number 100000 appeared on the screen. How many times has a number with the last digit 0 appeared on the screen?

Alice and Bob are playing the following game: They take turns writing on the board natural numbers not exceeding $2018$ (to write the number twice is forbidden). Alice begins. A player wins if after his or her move there appear three numbers on the board which are in arithmetic progression. Which player has a winning strategy?

$n$ men and one woman are in the meeting room with $n+1$ chairs, each of them having their own seat. Show that the following two number of cases are equal. (1) Number of cases to choose one man to get out of the room, and make the left $n-1$ men to sit to each other's chair. (2) Number of cases to make $n+1$ people to sit to each other's chair.

Six teams participate in a hockey tournament. Each team plays once against every other team. In each game, a team is awarded $3$ points for a win, $1$ point for a draw, and none for a loss. After the tournament the teams are ranked by total points. No two teams have the same total points. Each team (except the bottom team) has $2$ points more than the team ranking one place lower. Prove that the team that finished fourth has won two games and lost three games.

Let $S$ be a set of $n\ge2$ positive integers. Prove that there exist at least $n^2$ integers that can be written in the form $x+yz$ with $x,y,z\in S$. [i]Proposed by Gerhard Woeginger, Austria[/i]

a) Prove that you can rearrange the numbers $1, 2, ... , 32$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean. b) Can you rearrange the numbers $1, 2, ... , 100$ in such a way, that for every couple of numbers none of the numbers between them will equal their arithmetic mean?

On a river with a current speed of \(3 \, \text{km/h}\), there are two harbors. Every Saturday, a cruise ship departs from Harbor 1 to Harbor 2, stays overnight, and returns to Harbor 1 on Sunday. On the ship live two snails, Romeo and Juliet. One Saturday, immediately after the ship departs, both snails start moving to meet each other and do so exactly when the ship arrives at Harbor 2. On the following Sunday, as the ship departs from Harbor 2, Romeo starts moving towards Juliet's house and reaches there exactly when the ship arrives back at Harbor 1. Given that Juliet moves half as fast as Romeo, determine the speed of the ship in still water. [i]Proposed by Demetre Gelashvili, Georgia [/i]

p1. Known points $A (-1.2)$, $B (0,2)$, $C (3,0)$, and $D (3, -1)$ as seen in the following picture. Determine the measure of the angle $AOD$ . [img]https://cdn.artofproblemsolving.com/attachments/f/2/ca857aaf54c803db34d8d52505ef9a80e7130f.png[/img] p2. Determine all prime numbers $p> 2$ until $p$ divides $71^2 - 37^2 - 51$. p3. A ball if dropped perpendicular to the ground from a height then it will bounce back perpendicular along the high third again, down back upright and bouncing back a third of its height, and next. If the distance traveled by the ball when it touches the ground the fourth time is equal to $106$ meters. From what height is the ball was dropped? p4. The beam $ABCD.EFGH$ is obtained by pasting two unit cubes $ABCD.PQRS$ and $PQRS.EFGH$. The point K is the midpoint of the edge $AB$, while the point $L$ is the midpoint of the edge $SH$. What is the length of the line segment $KL$? p5. How many integer numbers are no greater than $2004$, with remainder $1$ when divided by $2$, with remainder $2$ when divided by $3$, with remainder $3$ when divided by $4$, and with remainder $4$ when divided by $5$?

Gonçalo writes in a board four of the the following numbers $0, 1, 2, 3, 4$, he can repeat numbers. Nicolas can realize the following operation: change one number of the board, by the remainder(in the division by $5$) of the product of others two numbers of the board. Nicolas wins if all the four numbers are equal, determine if Gonçalo can choose numbers such that Nicolas will never win.

Let $n $ be a positive integer. Let $f $ be a function from nonnegative integers to themselves. Let $f (0,i)=f (i,0)=0$, $f (1, 1)=n $, and $ f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] $ for positive integers $i, j$ such that $i*j>1$. Find the number of pairs $(i,j) $ such that $f (i, j) $ is an odd number.( $[x]$ is the floor function).

Given six three-element subsets of a finite set $X$, show that it is possible to color the elements of $X$ in two colors so that none of the given subsets is in one color

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 $A(1), A(2),...,A(2014)$.

Madoka chooses $4$ random numbers $a, b, c, d$ between $0$ and $1$. She notices that $a+b+c = 1$. If the probability that $d > a, b, c$ can be written in simplest form as $\frac{m}{n}$, find $m + n$.

A natural number $n$ is written on the board. Ben plays a game as follows: in every step, he deletes the number written on the board, and writes either the number which is three greater or two less than the number he has deleted. Is it possible that for every value of $n$, at some time, he will get to the number $2020$?