Determine the number of $n$-tuples of integers $(x_1,x_2,\cdots ,x_n)$ such that $|x_i| \le 10$ for each $1\le i \le n$ and $|x_i-x_j| \le 10$ for $1 \le i,j \le n$.

Prove that one can always mark $50$ points inside of any convex $100$-gon, so that each its vertix is on a straight line connecting some two marked points. (4)

Let $n \ge 2$. Ana and Beto play the following game: Ana chooses $2n$ non-negative real numbers $x_1, x_2,\ldots , x_{2n}$ (not necessarily different) whose total sum is $1$, and shows them to Beto. Then Beto arranges these numbers in a circle in the way she sees fit, calculates the product of each pair of adjacent numbers, and writes the maximum value of these products. Ana wants to maximize the number written by Beto, while Beto wants to minimize it. What number will be written if both play optimally?

In a matrix $2n \times 2n$, $n \in N$, are $4n^2$ real numbers with a sum equal zero. The absolute value of each of these numbers is not greater than $1$. Prove that the absolute value of a sum of all the numbers from one column or a row doesn't exceed $n$.

There are $24$ robots on the plane. Each robot has a $70^{\circ}$ field of view. What is the maximum number of observing relations? (Observing is a one-sided relation)

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

Let $n$ be a positive integer, and let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word, consisting of just letters $a$ and/or $b$. Suppose that the minimal period $N$ of $W$ is greater than $2^n$. A finite nonempty word $U$ is said to [i]appear[/i] in $W$ if there exist indices $k \leq \ell$ such that $U=x_k x_{k+1} \ldots x_{\ell}$. A finite word $U$ is called [i]ubiquitous[/i] if the four words $Ua$, $Ub$, $aU$, and $bU$ all appear in $W$. Prove that there are at least $n$ ubiquitous finite nonempty words. [i]Proposed by Grigory Chelnokov, Russia[/i]

Consider a cross like in the figure but with size $2021$. Every square have a $+1$. Every minute we select a sub-cross of size $3$ and multiply their squares by $-1$. It´s posible achieve that all the squares of the cross with size $2021$ have a $-1$?

For a binary string $S$ (i.e. a string of 0 's and 1's) that contains at least one 0 , we produce a binary string $f(S)$ as follows: - If the substring 110 occurs in $S$, replace each instance of 110 with 01 to produce $f(S)$; - Otherwise, replace the leftmost occurrence of 0 in $S$ by 1 to produce $f(S)$. Given binary string $S$ of length $n$, we define the lifetime of $S$ to be the number of times $f$ can be applied to $S$ until the resulting string contains no more 0 's. For example, $$

$n$ parallel segments of lengths $a_1 \le a_2 \le a_3 \le ... \le a_n$ were painted to mark an airport atrium. However, the architect decided that the $n$ segments should have equal length. If the cost per meter of extending the lines is equal to the cost of reducing them, how long should the lines be in order to minimize costs?

The beaver is chess piece that move to $2$ cells by horizontal or vertical. Every cell of $100 \times 100$ chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need?

There are $ n \plus{} 1$ cells in a row labeled from $ 0$ to $ n$ and $ n \plus{} 1$ cards labeled from $ 0$ to $ n$. The cards are arbitrarily placed in the cells, one per cell. The objective is to get card $ i$ into cell $ i$ for each $ i$. The allowed move is to find the smallest $ h$ such that cell $ h$ has a card with a label $ k > h$, pick up that card, slide the cards in cells $ h \plus{} 1$, $ h \plus{} 2$, ... , $ k$ one cell to the left and to place card $ k$ in cell $ k$. Show that at most $ 2^n \minus{} 1$ moves are required to get every card into the correct cell and that there is a unique starting position which requires $ 2^n \minus{} 1$ moves. [For example, if $ n \equal{} 2$ and the initial position is 210, then we get 102, then 012, a total of 2 moves.]

At a party there are eight guests, and each participant can't talk with at most three persons. Prove that we can group the persons in four pairs such that in every pair a conversation can take place.

A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of $2\sqrt{2}$ from its original location? [i]Proposed by Connor Gordon[/i]

Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has.

[b]p1.[/b] A triangle $ABC$ is drawn in the plane. A point $D$ is chosen inside the triangle. Show that the sum of distances $AD+BD+CD$ is less than the perimeter of the triangle. [b]p2.[/b] In a triangle $ABC$ the bisector of the angle $C$ intersects the side $AB$ at $M$, and the bisector of the angle $A$ intersects $CM$ at the point $T$. Suppose that the segments $CM$ and $AT$ divided the triangle $ABC$ into three isosceles triangles. Find the angles of the triangle $ABC$. [b]p3.[/b] You are given $100$ weights of masses $1, 2, 3,..., 99, 100$. Can one distribute them into $10$ piles having the following property: the heavier the pile, the fewer weights it contains? [b]p4.[/b] Each cell of a $10\times 10$ table contains a number. In each line the greatest number (or one of the largest, if more than one) is underscored, and in each column the smallest (or one of the smallest) is also underscored. It turned out that all of the underscored numbers are underscored exactly twice. Prove that all numbers stored in the table are equal to each other. [b]p5.[/b] Two stores have warehouses in which wheat is stored. There are $16$ more tons of wheat in the first warehouse than in the second. Every night exactly at midnight the owner of each store steals from his rival, taking a quarter of the wheat in his rival's warehouse and dragging it to his own. After $10$ days, the thieves are caught. Which warehouse has more wheat at this point and by how much? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Alice and Bob play a game on a Cartesian Coordinate Plane. At the beginning, Alice chooses a lattice point $ \left(x_{0}, y_{0}\right) $ and places a pudding. Then they plays by turns (B goes first) according to the rules a. If $ A $ places a pudding on $ \left(x,y\right) $ in the last round, then $ B $ can only place a pudding on one of $ \left(x+2, y+1\right), \left(x+2, y-1\right), \left(x-2, y+1\right), \left(x-2, y-1\right) $ b. If $ B $ places a pudding on $ \left(x,y\right) $ in the last round, then $ A $ can only place a pudding on one of $ \left(x+1, y+2\right), \left(x+1, y-2\right), \left(x-1, y+2\right), \left(x-1, y-2\right) $ Furthermore, if there is already a pudding on $ \left(a,b\right) $, then no one can place a pudding on $ \left(c,d\right) $ where $ c \equiv a \pmod{n}, d \equiv b \pmod{n} $. 1. Who has a winning strategy when $ n = 2018 $ 1. Who has a winning strategy when $ n = 2019 $

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

A cunning dragon guards a princess. To overcome the dragon and to win the princess you must solve the following task: The dragon has in some of the fields $i$ the columned hall (see figure) with the numbers $1-8$. Even in the rest of the fields you can place numbers $9-36$. The numbers $1-36$ must be arranged so that any turn that starts with one enters from either the south or the west, and ends up going out towards either the north or east, goes through at least one number from the $5$ table. (On the figure are north, south, east and west indicated by N, S, E and W). Georg wants to win the princess. Is it possible to be done? [img]https://cdn.artofproblemsolving.com/attachments/0/7/2ad1b7f944847ee6d3c614ea6c2656865808e7.png[/img]

Two players play the following game on a square of $N \times N$ squares. They color one square in turn so that no two colored squares are on the same diagonal. A player who cannot make a move loses. For what values of $N$ does the first player have a winning strategy?

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

A $8\times 8$ board is divided in dominoes (rectangles with dimensions $1 \times 2$ or $2 \times 1$). [list=a] [*] Prove that the total length of the border between horizontal and vertical dominoes is at most $52$. [*] Determine the maximum possible total length of the border between horizontal and vertical dominoes. [/list] [i]Proposed by B. Frenkin, A. Zaslavsky, E. Arzhantseva[/i]

In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a $\tfrac{2}{3}$ probability of voting for Alice and a $\tfrac{1}{3}$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?

One hundred empty glasses are arranged in a $10 \times 10$ array. Now we pick $a$ of the rows and pour blue liquid into all glasses in these rows, so that they are half full. The remaining rows are filled halfway with yellow liquid. Afterwards, we pick $b$ of the columns and fill them up with blue liquid. The remaining columns are filled up with yellow liquid. The mixture of blue and yellow liquid turns green. If both halves have the same colour, then that colour remains as it is. [list=a] [*] Determine all possible combinations of values for $a$ and $b$ so that exactly half of the glasses contain green liquid at the end. [*] Is it possible that precisely one quarter of the glasses contain green liquid at the end? [/list]

Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational