Alice and Bob are playing the following game. Each turn Alice suggests an integer and Bob writes down either that number or the sum of that number with all previously written numbers. Is it always possible for Alice to ensure that at some moment among the written numbers there are [list=a] [*]at least a hundred copies of number 5? [*]at least a hundred copies of number 10? [/list] [i]Andrey Arzhantsev[/i]

Some unit squares of $ 2007\times 2007 $ square board are colored. Let $ (i,j) $ be a unit square belonging to the $ith$ line and $jth$ column and $ S_{i,j} $ be the set of all colored unit squares $(x,y)$ satisfying $ x\leq i, y\leq j $. At the first step in each colored unit square $(i,j)$ we write the number of colored unit squares in $ S_{i,j} $ . In each step, in each colored unit square $(i,j)$ we write the sum of all numbers written in $ S_{i,j} $ in the previous step. Prove that after finite number of steps, all numbers in the colored unit squares will be odd.

A [i]labyrinth[/i] is a system of $2024$ caves and $2023$ non-intersecting (bidirectional) corridors, each of which connects exactly two caves, where each pair of caves is connected through some sequence of corridors. Initially, Erik is standing in a corridor connecting some two caves. In a move, he can walk through one of the caves to another corridor that connects that cave to a third cave. However, when doing so, the corridor he was just in will magically disappear and get replaced by a new one connecting the end of his new corridor to the beginning of his old one (i.e., if Erik was in a corridor connecting caves $a$ and $b$ and he walked through cave $b$ into a corridor that connects caves $b$ and $c$, then the corridor between caves $a$ and $b$ will disappear and a new corridor between caves $a$ and $c$ will appear). Since Erik likes designing labyrinths and has a specific layout in mind for his next one, he is wondering whether he can transform the labyrinth into that layout using these moves. Prove that this is in fact possible, regardless of the original layout and his starting position there.

On an international conference there are 4 official languages. Each two of the attendees can have a conversation on one of the languages. Prove that at least 60% of the attendees can speak the same language.

At a conference, six people place their name badges in a hat, which is shaken up; one badge is then distributed to each person such that each distribution is equally likely. Each turn, every person who does not yet have their own badge finds the person whose badge they have and takes that person's badge. For example, if Alice has Bob's badge and Bob has Charlie's badge, Alice would have Charlie's badge after a turn. Compute the probability that everyone will eventually end up with their own badge.

There are $n$ passengers in a line, waiting to board a plane with $n$ seats. For $1 \le k \le n$, the $k^{th}$ passenger in line has a ticket for the $k^{th}$ seat. However, the rst passenger ignores his ticket, and decides to sit in a seat at random. Thereafter, each passenger sits as follows: If his/her assigned is empty, then he/she sits in it. Otherwise, he/she sits in an empty seat at random. How many different ways can all $n$ passengers be seated?

[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$. [b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials. [b]p3.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p4.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? [b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a board consisting of $n\times n$ unit squares where $n \ge 2$. Two cells are called neighbors if they share a horizontal or vertical border. In the beginning, all cells together contain $k$ tokens. Each cell may contain one or several tokens or none. In each turn, choose one of the cells that contains at least one token for each of its neighbors and move one of those to each of its neighbors. The game ends if no such cell exists. (a) Find the minimal $k$ such that the game does not end for any starting configuration and choice of cells during the game. (b) Find the maximal $k$ such that the game ends for any starting configuration and choice of cells during the game. Proposed by Theresia Eisenkölbl

(I'll skip over the whole "dressing" of the graph in cities and flights [color=#FF0000][Mod edit: Shu has posted the "dressed-up" version below][/color]) For an ordinary directed graph, show that there is a subset A of vertices such that: $1.$ There are no edges between the vertices of A. $2.$ For any vertex $v$, there is either a direct way from $v$ to a vertex in A, or a way passing through only one vertex and ending in A (like $v$ ->$v'$-> $a$, where $a$ is a vertex in A)

On a board the numbers $1,2,3,\dots,98,99$ are written. One has to mark $50$ of them, such that the sum of two marked numbers is never equal to $99$ or $100$. How many ways one can mark these numbers?

Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David? [i]Proposed by Vlad Spătaru[/i]

Jeck and Lisa are playing a game on table dimensions $m \times n$ , where $m , n >2$. Lisa starts so that she puts knight figurine on arbitrary square of table.After that Jeck and Lisa put new figurine on table by the following rules: [b]1.[/b] Jeck puts queen figurine on any empty square of a table which is two squares vertically and one square horizontally distant, or one square vertically and two squares horizontally distant from last knight figurine which Lisa put on the table [b]2.[/b] Lisa puts knight figurine on any empty square of a table which is in the same row, column or diagonal as last queen figurine Jeck put on the table. Player which cannot put his figurine loses. For which pairs of $(m,n)$ Lisa has winning strategy? [i] Proposed by Stijn Cambie[/i]

Let $B_n$ be the set of all sequences of length $n$, consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$ \varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1). $$. Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $. .

A mathematical frog jumps along the number line. The frog starts at $1$, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n + 2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n.$ Show that if $k \geq 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$

The $14 \times 14$ chessboard squares are colored in pattern, as shown in the picture. Can you choose seven fields blacks and seven white squares of this chessboard in such a way, that there is exactly one selected field in each row and column? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8ba46030cd0f0e0511f1f9e723e5bd29e9975.png[/img]

On a blackboard, there are $11$ positive integers. Show that one can choose some (maybe all) of these numbers and place "$+$" and "$-$" in between such that the result is divisible by $2011$.

Let $m$ and $n$ be positive integers. Show that $\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$

Given a square board with dimensions $1 995 \times 1 995$. These cells are painted with black and white paints in checkerboard order like this. that the corner cells are black. Two black and one white cells were randomly cut out of the board. Prove that the rest of the board can be divided into rectangles of size $1 \times 2$ .

A $7\times 7$ board has a lamp on each of its $49$ squares, which can be on or off. The allowed operation is to choose $3$ consecutive cells of a row or a column that have two lamps neighboring each other on and the other off, and change the state of all three. Namely [img]https://cdn.artofproblemsolving.com/attachments/e/b/28737b19c940ff5e1c98d05533c77069e990f5.png[/img] Give a configuration of exactly $8$ lit lamps located in the first $4$ rows of the board such that, through a succession of permitted operations, a single lamp is lit on the board and that it is located in the last row. Show the sequence of operations used to achieve the goal.

A piece of rectangular paper $20 \times 19$, divided into four units, is cut into several square pieces, the cuts being along the sides of the unit squares. Such a square piece is called odd square if the length of its side is an odd number. a) What is the minimum possible number of odd squares? b) What is the smallest value that the sum of the perimeters of the odd squares can take?

Let $x$ be an irrational number between 0 and 1 and $x = 0.a_1a_2a_3\cdots$ its decimal representation. For each $k \ge 1$, let $p(k)$ denote the number of distinct sequences $a_{j+1} a_{j+2} \cdots a_{j+k}$ of $k$ consecutive digits in the decimal representation of $x$. Prove that $p(k) \ge k+1$ for every positive integer $k$.

Let $a_k$ be the number of ordered $10$-tuples $(x_1, x_2, ..., x_{10})$ of nonnegative integers such that $$x^2_1+ x^2_2+ ... + x^2_{10} = k.$$ Let $b_k = 0$ if $a_k$ is even and $b_k = 1$ if $a_k$ is odd. Find $\sum^{2012}_{i=1} b_{4i}$.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Given positive integers $a,b,$ find the least positive integer $m$ such that among any $m$ distinct integers in the interval $[-a,b]$ there are three pair-wise distinct numbers that their sum is zero. [i]Proposed by Marian Tetiva, Romania[/i]

For which $n \in N$ is it possible to arrange a tennis tournament for doubles with $n$ players such that each player has every other player as an opponent exactly once?