[u]Round 1[/u] [b]p1.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p2.[/b] A positive number is placed on each of the $10$ circles in this picture. It turns out that for each of the nine little equilateral triangles, the number on one of its corners is the sum of the numbers on the other two corners. Is it possible that all $10$ numbers are different? [img]https://cdn.artofproblemsolving.com/attachments/b/f/c501362211d1c2a577e718d2b1ed1f1eb77af1.png[/img] [b]p3.[/b] Pablo and Nina take turns entering integers into the cells of a $3 \times 3$ table. Pablo goes first. The person who fills the last empty cell in a row must make the numbers in that row add to $0$. Can Nina ensure at least two of the columns have a negative sum, no matter what Pablo does? [b]p4. [/b]All possible simplified fractions greater than $0$ and less than $1$ with denominators less than or equal to $100$ are written in a row with a space before each number (including the first). Zeke and Qing play a game, taking turns choosing a blank space and writing a “$+$” or “$-$” sign in it. Zeke goes first. After all the spaces have been filled, Zeke wins if the value of the resulting expression is an integer. Can Zeke win no matter what Qing does? [img]https://cdn.artofproblemsolving.com/attachments/3/6/15484835686fbc2aa092e8afc6f11cd1d1fb88.png[/img] [b]p5.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/0/c/d827cf26c8eaabfd5b0deb92612a6e6ebffb47.png[/img] [u]Round 2[/u] [b]p6.[/b] Prove that among any $3^{2022}$ integers, it is possible to find exactly $3^{2021}$ of them whose sum is divisible by $3^{2021}$. [b]p7.[/b] Given a list of three numbers, a zap consists of picking two of the numbers and decreasing each of them by their average. For example, if the list is $(5, 7, 10)$ and you zap $5$ and $10$, whose average is $7.5$, the new list is $(-2.5, 7, 2.5)$. Is it possible to start with the list $(3, 1, 4)$ and, through some sequence of zaps, end with a list in which the sum of the three numbers is $0$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Jerry writes down all binary strings of length $10$ without any two consecutive $1$s. How many $1$s does Jerry write?

In a circle with a radius of $1$, $64$ mutually different points are selected. Prove that $10$ mutually different points can be selected from them, which lie in a circle with a radius $\frac12$.

A triangle is constructed with the lengths of the sides chosen from the set $\{2, 3, 5, 8, 13, 21, 34, 55, 89, 144\}$. Show that this triangle must be isosceles. (A triangle is isosceles if it has at least two sides the same length.)

Hannibal and Clarice are still at a barbecue and there are three anticuchos left, each of which it has $10$ pieces. Of the $30$ total pieces, there are $29$ chicken and one meat, the which is at the bottom of one of the anticuchos. To decide who to stay with the piece of meat, they decide to play the following game: they alternately take out a piece of one of the anticuchos (they can take only the outer pieces) and whoever wins the game manages to remove the piece of meat. Clarice decides if she starts or Hannibal starts. What should she decide?

An integer $n\ge2$ is called [i]amicable[/i] if there exists subsets $A_1,A_2,\ldots,A_n$ of the set $\{1,2,\ldots,n\}$ such that (i) $i\notin A_i$ for any $i=1,2,\ldots,n$, (ii) $i\in A_j$ for any $j\notin A_i$, for any $i\ne j$ (iii) $A_i\cap A_j\ne\emptyset$ for any $i,j\in\{1,2,\ldots,n\}$ (a) Prove that $7$ is amicable. (b) Prove that $n$ is amicable if and only if $n\ge7$.

In a game, the first player paints a point on the plane red; the second player paints 10 uncoloured points on the plane green; then the first player paints an uncoloured point on the plane red; the second player paints 10 uncoloured points on the plane green; and so on. The first player wins if there are three red points which form an equilateral triangle. Can the second player prevent the first player from winning? (A Kanel)

On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet. After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls. Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L},$ Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n,$ what are all possible values for $k(\mathfrak{L}),$ where $\mathfrak{L}$ is a labyrinth with $n$ walls?

An investigator works out that he needs to ask at most $91$ questions on the basis that all the answers will be yes or no and all will be true. The questions may depend upon the earlier answers. Show that he can make do with $105$ questions if at most one answer could be a lie.

Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$, where $n$ is a positive integer.

There are $n$ cities on each side of Hung river, with two-way ferry routes between some pairs of cities across the river. A city is “convenient” if and only if the city has ferry routes to all cities on the other side. The river is “clear” if we can find $n$ different routes so that the end points of all these routes include all $2n$ cities. It is known that Hung river is currently unclear, but if we add any new route, then the river becomes clear. Determine all possible values for the number of convenient cities. [i] Proposed by usjl[/i]

Let $A$ be a set contains $2000$ distinct integers and $B$ be a set contains $2016$ distinct integers. $K$ is the numbers of pairs $(m,n)$ satisfying \[ \begin{cases} m\in A, n\in B\\ |m-n|\leq 1000 \end{cases} \] Find the maximum value of $K$

During a game of chess, at a certain point, in each row and column of the board there is an odd number of pieces. Prove that the number of pieces that are on black squares is even. (Note: a chessboard has $8$ rows and $8$ columns)

A convex $n$-gon $A_0A_1\dots A_{n-1}$ has been partitioned into $n-2$ triangles by certain diagonals not intersecting inside the $n$-gon. Prove that these triangles can be labeled $\triangle_1,\triangle_2,\dots,\triangle_{n-2}$ in such a way that $A_i$ is a vertex of $\triangle_i$, for $i=1,2,\dots,n-2$. Find the number of all such labellings.

An [i]$n$-folding process[/i] on a rectangular piece of paper with sides aligned vertically and horizontally consists of repeating the following process $n$ times: [list] [*]Take the piece of paper and fold it in half vertically (choosing to either fold the right side over the left, or the left side over the right). [*]Rotate the paper $90^\circ$ clockwise. [/list] A $10$-folding process is performed on a piece of paper, resulting in a $1$-by-$1$ square base consisting of many stacked layers of paper. Let $d(i,j)$ be the Euclidean distance between the center of the $i$th square from the top and the center of the $j$th square from the top when the paper is unfolded. Determine the maximum possible value of $\sum_{i=1}^{1023} d(i, i+1).$

Let $n$ be a positive integer. A square $ABCD$ is partitioned into $n^2$ unit squares. Each of them is divided into two triangles by the diagonal parallel to $BD$. Some of the vertices of the unit squares are colored red in such a way that each of these $2n^2$ triangles contains at least one red vertex. Find the least number of red vertices. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 4)[/i]

In a square field of side length $12$ there is a source that contains a system of straight irrigation ditches. This is laid out in such a way that for every point of the field the distance to the next ditch is at most $1$. Here, the source is as a point and are the ditches to be regarded as stretches. It must be verified that the total length of the irrigation ditches is greater than $70$ m. The sketch shows an example of a trench system of the type indicated. [img]https://cdn.artofproblemsolving.com/attachments/6/5/5b51511da468cf14b5823c6acda3c4d2fe8280.png[/img]

[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$ [i] Ivan Landgev[/i]

The expression $ x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?

Six numbers are placed on a circle. For every number $A$ we have: $A$ equals the absolute value of $(B- C)$, where $B$ and $C$ follow $A$ clockwise. The total sum of the numbers equals $1$. Find all the numbers. (Folklore)

You plan to organize your birthday party, which will be attended either by exactly $m$ persons or by exactly $n$ persons (you are not sure at the moment). You have a big birthday cake and you want to divide it into several parts (not necessarily equal), so that you are able to distribute the whole cake among the people attending the party with everybody getting cake of equal mass (however, one may get one big slice, while others several small slices - the sizes of slices may differ). What is the minimal number of parts you need to divide the cake, so that it is possible, regardless of the number of guests.

Determine the sets $S{}$ of positive integers satisfying the following two conditions: [list=a] [*]For any positive integers $a, b, c{}$, if $ab + bc + ca{}$ is in $S$, then so are $a + b + c{}$ and $abc$; and [*]The set $S{}$ contains an integer $N \geqslant 160$ such that $N-2$ is not divisible by $4$. [/list] [i]Bogdan Blaga, United Kingdom[/i]

Several fleas sit on the squares of a $10\times 10$ chessboard (at most one fea per square). Every minute, all fleas simultaneously jump to adjacent squares. Each fea begins jumping in one of four directions (up, down, left, right), and keeps jumping in this direction while it is possible; otherwise, it reverses direction on the opposite. It happened that during one hour, no two fleas ever occupied the same square. Find the maximal possible number of fleas on the board.

Given $n\ge 3$. consider a sequence $a_1,a_2,...,a_n$, if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$, we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.

A square grid $6 \times 6$ is tiled with $18$ dominoes. Prove that there is a line, intersecting no dominoes. Can this line be unique?