Prove that there are different points $A_0 \,\, ,A_1 \,\, , \cdots A_{2550}$ on the $XY$ plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If $ d(A_i,A_j)$ represents the distance between $A_i\,\, , A_j $ then $$ \sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008}$$ Note : $ \{x \}$ represents the decimal part of x e.g. $ \{ 3.16\} = 0.16$. [i] (passer-by)[/i]

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

Many people use the social network "BWM". It is known that: By choosing any four people of that network there always is one that is a friend of the other three. Is it then true that by choosing any four people there always is one that is a friend of everyone in "BWM"? [b]Note:[/b] If member $A$ is a friend of member $B$, then member $B$ also is a friend of member $A$.

There are more than $n^2$ stones on the table. Peter and Vasya play a game, Peter starts. Each turn, a player can take any prime number less than $n$ stones, or any multiple of $n$ stones, or $1$ stone. Prove that Peter always can take the last stone (regardless of Vasya's strategy). [i]S Berlov[/i]

$(a_{1}, a_{2}, ... , a_{2n})$ is a permutation of $\{1, 2, ... , 2n\}$ such that $|a_{i}-a_{i+1}| \neq |a_{j}-a_{j+1}|$ for $i \neq j$. Show that $a_{1}= a_{2n}+n$ iff $1 \leq a_{2i}\leq n$ for $i = 1, 2, ... n.$

$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that \[bc<2a^2<4bc\]

A game is played with two players and an initial stack of $n$ pennies $(n \geq 3)$. The players take turns choosing one of the stacks of pennies on the table and splitting it into two stacks. The winner is the player who makes a move that causes all stacks to be of height $1$ or $2.$ For which starting values of n does the player who goes first win, assuming best play by both players?

In a school, there are $1000$ students in each year level, from Year $1$ to Year $12$. The school has $12 000$ lockers, numbered from $1$ to $12 000$. The school principal requests that each student is assigned their own locker, so that the following condition is satisfied: For every pair of students in the same year level, the difference between their locker numbers must be divisible by their year-level number. Can the principal’s request be satisfied?

Diagonals of a $1$ by $1$ square are arranged in an $8$ by $8$ table (one in each $1$ by $1 $ square). Consider the union $W$ of all $64$ diagonals drawn. The set $W$ consists of several connected pieces (two points belong to the same piece if and only if W contains a path between them). Can the number of the pieces be greater than (a) $15$, (b) $20$? (NB Vassiliev)

Sone of vertices of the infinite grid $\mathbb{Z}^{2}$ are missing. Let's take the remainder as a graph. Connect two edges of the graph if they are the same in one component and their other components have a difference equal to one. Call every connected component of this graph a [b]branch[/b]. Suppose that for every natural $n$ the number of missing vertices in the $(2n+1)\times(2n+1)$ square centered by the origin is less than $\frac{n}{2}$. Prove that among the branches of the graph, exactly one has an infinite number of vertices. Time allowed for this problem was 90 minutes.

The digits $0, 1 , 2, ..., 9$ are written in a $10 x 10$ table , each number appearing $10$ times . (a) Is it possible to write them in such a way that in any row or column there would be not more than $4$ different digits? (b) Prove that there must be a row or column containing more than $3$ different digits . { L . D . Kurlyandchik , Leningrad)

Given a board $3 \times 2021$, all cells of which are white. Two players in turns colour two white cells, which are either in the same row or column, in black. A player, which can not make a move, loses. Which of the player can guarantee his win regardless of the moves of his opponent?

Is it possible to fill the cells of a table of size $2019\times2019$ with pairwise distinct positive integers in such a way that in each rectangle of size $1\times2$ or $2\times1$ the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most $2019^4?$

In a board school, there are 9 subjects, 512 students, and 256 rooms (two people in each room.) For every student there is a set (a subset of the 9 subjects) of subjects the student is interested in. Each student has a different set of subjects, (s)he is interested in, from all other students. (Exactly one student has no subjects (s)he is interested in.) Prove that the whole school can line up in a circle in such a way that every pair of the roommates has the two people standing next to each other, and those pairs of students standing next to each other that are not roommates, have the following properties. One of the two students is interested in all the subjects that the other student is interested in, and also exactly one more subject.

[u]Round 1[/u] [b]p1.[/b] What is $2 + 22 + 1 + 3 - 31 - 3$? [b]p2.[/b] Let $ABCD$ be a rhombus. Given $AB = 5$, $AC = 8$, and $BD = 6$, what is the perimeter of the rhombus? [b]p3.[/b] There are $2$ hats on a table. The first hat has $3$ red marbles and 1 blue marble. The second hat has $2$ red marbles and $4$ blue marbles. Jordan picks one of the hats randomly, and then randomly chooses a marble from that hat. What is the probability that she chooses a blue marble? [u]Round 2[/u] [b]p4.[/b] There are twelve students seated around a circular table. Each of them has a slip of paper that they may choose to pass to either their clockwise or counterclockwise neighbor. After each person has transferred their slip of paper once, the teacher observes that no two students exchanged papers. In how many ways could the students have transferred their slips of paper? [b]p5.[/b] Chad wants to test David's mathematical ability by having him perform a series of arithmetic operations at lightning-speed. He starts with the number of cubic centimeters of silicon in his 3D printer, which is $109$. He has David perform all of the following operations in series each second: $\bullet$ Double the number $\bullet$ Subtract $4$ from the number $\bullet$ Divide the number by $4$ $\bullet$ Subtract $5$ from the number $\bullet$ Double the number $\bullet$ Subtract $4$ from the number Chad instructs David to shout out after three seconds the result of three rounds of calculations. However, David computes too slowly and fails to give an answer in three seconds. What number should David have said to Chad? [b]p6.[/b] Points $D, E$, and $F$ lie on sides $BC$, $CA$, and $AB$ of triangle $ABC$, respectively, such that the following length conditions are true: $CD = AE = BF = 2$ and $BD = CE = AF = 4$. What is the area of triangle $ABC$? [u]Round 3[/u] [b]p7.[/b] In the $2, 3, 5, 7$ game, players count the positive integers, starting with $1$ and increasing, which do not contain the digits $2, 3, 5$, and $7$, and also are not divisible by the numbers $2, 3, 5$, and $7$. What is the fifth number counted? [b]p8.[/b] If A is a real number for which $19 \cdot A = \frac{2014!}{1! \cdot 2! \cdot 2013!}$ , what is $A$? Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ...\cdot 2 \cdot 1$. [b]p9.[/b] What is the smallest number that can be written as both $x^3 + y^2$ and $z^3 + w^2$ for positive integers $x, y, z,$ and $w$ with $x \ne z$? [u]Round 4[/u] [i]Each of the three problems in this round depends on the answer to one of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. In addition, it is given that the answer to each of the following problems is a positive integer less than or equal to the problem number. [/i] [b]p10.[/b] Let $B$ be the answer to problem $11$ and let $C$ be the answer to problem $12$. What is the sum of a side length of a square with perimeter $B$ and a side length of a square with area $C$? [b]p11.[/b] Let $A$ be the answer to problem $10$ and let $C$ be the answer to problem $12$. What is $(C - 1)(A + 1) - (C + 1)(A - 1)$? [b]p12.[/b] Let $A$ be the answer to problem $10$ and let $B$ be the answer to problem $11$. Let $x$ denote the positive difference between $A$ and $B$. What is the sum of the digits of the positive integer $9x$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2915810p26040675]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.

A [i]lattice point[/i] in the plane is a point with integer coordinates. Let $T$ be a triangle in the plane whose vertice are lattice points, but with no other lattice points on its sides. Furthermore, suppose $T$ contains exactly four lattice points in its interior. Prove that these four points lie on a straight line.

There are $40$ points on the two parallel lines. We divide it to pairs, such that line segments, that connects point in pair, do not intersect each other ( endpoint from one segment cannot lies on another segment). Prove, that number of ways to do it is less than $3^{39}$

Let $n$ be a positive integer greater than $1$ and let us call an arbitrary set of cells in a $n\times n$ square $\textit{good}$ if they are the intersection cells of several rows and several columns, such that none of those cells lie on the main diagonal. What is the minimum number of pairwise disjoint $\textit{good}$ sets required to cover the entire table without the main diagonal?

There are $n$ boys and $n$ girls sitting around a circular table, where $n>3$. In every move, we are allowed to swap the places of $2$ adjacent children. The [b]entropy[/b] of a configuration is the minimal number of moves such that at the end of them each child has at least one neighbor of the same gender. Find the maximal possible entropy over the set of all configurations. [i]Authored by Viktor Simjanoski[/i]

A teacher is known to have $2^k$ apples for some $k \in \mathbb{N}$. He ets one of the apples and distributes the rest of the apples to his students $A$ and $B$. The students do not see how many apples the other gets, and they do not know the number $k$. However, they have pre-selected a discreet way to reveal one another something about the number of apples: each of the students scratches their head either by their right, left or both hands, depending on the number of apples they have received. To the teacher's surprise, the students will always know which one of the students got more apples, or that the teacher ate the only apple by herself. How is this possible?

$31$ guests at a party sit in equally spaced chairs around a round table , but they have not noticed that there are cards with the names of the guests on the stalls. (a) Assuming they have been so unlucky that no one is in the room which corresponds to him, show that it is possible to get at least two people to stay in their correct position, without anyone getting up from their seat, turning the table. (b) Show a configuration where exactly one guest is in his assigned place and where in no way that the table is turned it is possible to achieve that at least two remain right.

A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?

Let $C$ be a coloring of all edges and diagonals of a convex $n$−gon in red and blue (in Romanian, rosu and albastru). Denote by $q_r(C)$ (resp. $q_a(C)$) the number of quadrilaterals having all its edges and diagonals red (resp. blue). Prove: $ \underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}$

Kolya is told that two of his four coins are fake. He knows that all real coins have the same weight, all fake coins have the same weight, and the weight of a real coin is greater than that of a fake coin. Can Kolya decide whether he indeed has exactly two fake coins by using a balance twice?