Michael is at the centre of a circle of radius $100$ metres. Each minute, he will announce the direction in which he will be moving. Catherine can leave it as is, or change it to the opposite direction. Then Michael moves exactly $1$ metre in the direction determined by Catherine. Does Michael have a strategy which guarantees that he can get out of the circle, even though Catherine will try to stop him?

The following grid represents a mountain range; the number in each cell represents the height of the mountain located there. Moving from a mountain of height $a$ to a mountain of height $b$ takes $(b - a)^2$ time. Suppose that you start on the mountain of height $1$ and that you can move up, down, left, or right to get from one mountain to the next. What is the minimum amount of time you need to get to the mountain of height $49$? [img]https://cdn.artofproblemsolving.com/attachments/0/6/10b07a2b2ae4ba750cfffc3dc678880333c2de.png[/img]

All $ 2n\ge 2 $ squares of a $ 2\times n $ rectangle are colored with three colors. We say that a color has a [i]cut[/i] if there is some column (from all $ n $) that has both squares colored with it. Determine: [b]a)[/b] the number of colorings that have no cuts. [b]b)[/b] the number of colorings that have a single cut.

Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.

Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same list of numbers that Daniel wrote from top to down. Find the greatest length of the Daniel's list can have.

Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.

There are 2016 points near a line such that the distance from each point to the line is less than 1 cm, and the distance between any two points is always greater than 2 cm. Prove that there exist two points whose distance is at least 17 meters.

The [i]triangular domino[/i] is a game that uses the tokens shown below, with equilateral triangle shape with side $ 1$. The idea of the game is to construct an equilateral triangle with side $n$, no gaps, following the rules of the domino or classic. $\bullet$ Show that the sum $S$ of the values corresponding to the edges that are part of the sides of the greater triangle, it depends only on n, and not on the way in which the tokens are paired. $\bullet$ For each value of $n$, calculate $S$. [img]https://cdn.artofproblemsolving.com/attachments/e/9/898664fac380725a7398dfe470298a90b8c69b.png[/img]

Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?

Let $P(A)$ be the arithmetic-means of all elements of set $A = \{ a_1, a_2, \ldots, a_n \}$, namely $P(A) = \frac{1}{n} \sum^{n}_{i=1}a_i$. We denote $B$ "balanced subset" of $A$, if $B$ is a non-empty subset of $A$ and $P(B) = P(A)$. Let set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$. Find the number of all "balanced subset" of $M$.

Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard. They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board. Prove that player $B$ can always prevent player $A$ from winning.

Non-negative integers are placed on the vertices of a $100$-gon, the sum of the numbers is $99$. Every minute at one of the vertices that is equal to $0$ will be replaced by $2$ and both its neighboring numbers are subtracted by $1$. Prove that after a while a negative number will appear on the board.

A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks are available. Find all positive integers $n$ such that it is possible to give each of the directors a set of keys to $n$ different locks, according to the requirements and regulations of the company.

Find all positive integers $ n $ with the following property: It is possible to fill a $ n \times n $ chessboard with one of arrows $ \uparrow, \downarrow, \leftarrow, \rightarrow $ such that 1. Start from any grid, if we follows the arrows, then we will eventually go back to the start point. 2. For every row, except the first and the last, the number of $ \uparrow $ and the number of $ \downarrow $ are the same. 3. For every column, except the first and the last, the number of $ \leftarrow $ and the number of $ \rightarrow $ are the same.

Consider a $2\times2$ square with one corner removed from $1\times1$ , leaving a shape in the form of $L$ . [asy] unitsize(0.5 cm); draw((1,0)--(1,2)--(0,2)--(0,0)--(2,0)--(2,1)--(0,1)); [/asy] We will call this figure [i]triomino[/i]. Determine all values of $m, n$ such that a rectangle of $m\times n$ can be exactly covered with triominos.

People from n different countries sit at a round table. Assume that for every two members of the same country their neighbours sitting next to them on the right hand side are from different countries. Find the largest possible number of people sitting around the table?

Let $T$ denote the $15$-element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.

In an $8 \times 8$ grid of squares, each square was colored black or white so that no $2\times 2$ square has all its squares in the same color. A sequence of distinct squares $x_1,\dots, x_m$ is called a [b]snake of length $m$[/b] if for each $1\leq i <m$ the squares $x_i, x_{i+1}$ are adjacent and are of different colors. What is the maximum $m$ for which there must exist a snake of length $m$?

Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers: \[|A+qA|\ge (q+1)|A|-C_q.\] [i]Proposed by Antal Balog.[/i]

A certain island has some ports along the coast and some towns inland. All roads on this island are one-way, and they do not meet except at a port or a town. Moreover, once you leave a certain port or town by road, there is no way you can return there by road. For any two ports $i$ and $j$, let $f_{ij}$ denote the number of different routes along the roads between $i$ and $j$. (a) Suppose there are four ports on the island: $1, 2, 3$ and $4$, in clockwise order. Show that $$f_{14}f_{23} \ge f_{13}f_{24}$$ (b) Suppose there were six ports on the island: $1, 2, 3, 4, 5$ and $6$, in clockwise order. Show that $$f_{16}f_{25}f_{34} + f_{15}f_{24}f_{36} + f_{14}f_{26}f_{35}\ge f_{16}f_{24}f_{35}+ f_{15}f_{26}f_{34} + f_{14}f_{25}f_{36}$$ (S Fomin}

Consider the string of length $ 6$ composed of three characters $ a$, $ b$, $ c$. For each string, if two $ a$s are next to each other, or two $ b$s are next to each other, then replace $ aa$ by $ b$, and replace $ bb$ by $ a$. Also, if $ a$ and $ b$ are next to each other, or two $ c$s are next to each other, remove all two of them (i.e. delete $ ab$, $ ba$, $ cc$). Determine the number of strings that can be reduced to $ c$, the string of length 1, by the reducing processes mentioned above.

In a school there are $ 10$ rooms. Each student from a room knows exactly one student from each one of the other $ 9$ rooms. Prove that the rooms have the same number of students (we suppose that if $ A$ knows $ B$ then $ B$ knows $ A$).

Let $A(n)$ be the least possible number of distinct points in the plane with the following property: For every $k = 1,2,...,n$ there is a line containing precisely $k$ of these points. Show that $A(n) =\left[\frac{n+1}{2}\right] \left[\frac{n+2}{2}\right]$

King Radford of Peiza is hosting a banquet in his palace. The King has an enormous circular table with $2021$ chairs around it. At The King's birthday celebration, he is sitting in his throne (one of the $2021$ chairs) and the other $2020$ chairs are filled with guests, with the shortest guest sitting to the King's left and the remaining guests seated in increasing order of height from there around the table. The King announces that everybody else must get up from their chairs, run around the table, and sit back down in some chair. After doing this, The King notices that the person seated to his left is different from the person who was previously seated to his left. Each other person at the table also notices that the person sitting to their left is different. Find a closed form expression for the number of ways the people could be sitting around the table at the end. You may use the notation $D_{n},$ the number of derangements of a set of size $n$, as part of your expression.

In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$. For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.