8 chess players participated in the chess tournament and everyone played exactly one game with everyone else. It is known that any two chess players who play a draw with each other ended up scoring different numbers of points. Find the greatest possible number of draws in this tournament. (For winning the game the chess player is awarded $1$ point, for a draw $1/2$ points, for defeat $0$ points.)

Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$. [i]Proposed by Aidan Duncan[/i]

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]

p1. Among the numbers $\frac15$ and $\frac14$ there are infinitely many fractional numbers. Find $999$ decimal numbers between $\frac15$ and $\frac14$ so that the difference between the next fractional number with the previous fraction constant. (i.e. If $x_1, x_2, x_3, x_4,..., x_{999}$ is a fraction that meant, then $x_2 - x_1= x_3 - x_3= ...= x_n - x_{n-1}=...=x_{999}-x_{998}$) p2. The pattern in the image below is: "Next image obtained by adding an image of a black equilateral triangle connecting midpoints of the sides of each white triangle that is left in the previous image." The pattern is continuous to infinity. [img]https://cdn.artofproblemsolving.com/attachments/e/f/81a6b4d20607c7508169c00391541248b8f31e.png[/img] It is known that the area of ​​the triangle in Figure $ 1$ is $ 1$ unit area. Find the total area of ​​the area formed by the black triangles in figure $5$. Also find the total area of the area formed by the black triangles in the $20$th figure. p3. For each pair of natural numbers $a$ and $b$, we define $a*b = ab + a - b$. The natural number $x$ is said to be the [i]constituent [/i] of the natural number $n$ if there is a natural number $y$ that satisfies $x*y = n$. For example, $2$ is a constituent of $6$ because there is a natural number 4 so that $2*4 = 2\cdot 4 + 2 - 4 = 8 + 2 - 4 = 6$. Find all constituent of $2005$. p4. Three people want to eat at a restaurant. To find who pays them to make a game. Each tossing one coin at a time. If the result is all heads or all tails, then they toss again. If not, then "odd person" (i.e. the person whose coin appears different from the two other's coins) who pay. Determine the number of all possible outcomes, if the game ends in tossing: a. First. b. Second. c. Third. d. Tenth. p5. Given the equation $x^2 + 3y^2 = n$, where $x$ and $y$ are integers. If $n < 20$ what number is $n$, and which is the respective pair $(x,y)$ ? Show that it is impossible to solve $x^2 + 3y^2 = 8$ in integers.

In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer.

A bug is walking on the surface of a Rubik's cube(cube $3 \times 3 \times 3$). It can go to the adjacent cell on the same face or on the adjacent face. One day the bug started walking from some cell and returned to it, and visited all other cells exactly once. Prove that he made an even amount of moves that changed the face he is on.

assume that X is a set of n number.and $0\leq k\leq n$.the maximum number of permutation which acting on $X$ st every two of them have at least k component in common,is $a_{n,k}$.and the maximum nuber of permutation st every two of them have at most k component in common,is $b_{n,k}$. a)proeve that :$a_{n,k}\cdot b_{n,k-1}\leq n!$ b)assume that p is prime number,determine the exact value of $a_{p,2}$.

Consider the numbers from $1$ to $16$. The "solitar" game consists in the arbitrary grouping of the numbers in pairs and replacing each pair with the great prime divisor of the sum of the two numbers (i.e from $(1,2); (3,4); (5,6);...;(15,16)$ the numbers which result are $3,7,11,5,19,23,3,31$). The next step follows from the same procedure and the games continues untill we obtain only one number. Which is the maximum numbers with which the game ends.

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/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?

Set $S$ to be a subset of size $68$ of $\{1,2,...,2015\}$. Prove that there exist $3$ pairwise disjoint, non-empty subsets $A,B,C$ such that $|A|=|B|=|C|$ and $\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c$

Kiko and Ñoño play with a rod of length $2n$ where $n \le 3$ is an integer. Kiko cuts the rod in $ k \le 2n$ pieces of integer lengths. Then Ñoño has to arrange these pieces so that they form a hexagon of equal opposite sides and equal angles. The pieces can not be split and they all have to be used. If Ñoño achieves his goal, he wins, in any other case, Kiko wins. Determine which victory can be secured based on $k$.

For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards. [i]Linus Tang[/i]

In the beginning, number $1$ has been written to point $(0,0)$ and $0$ has been written to any other point of integral coordinates. After every second, all numbers are replaced with the sum of the numbers in four neighbouring points at the previous second. Find the sum of numbers in all points of integral coordinates after $n$ seconds.

Along a track for cross-country skiing, $1000$ seats are placed in a row and numbered in order from $1$ to $1000$. By mistake, $n$ tickets were sold, $100 < n < 1000$, each with one of the numbers $1,2,..., 100$ printed on it. Also for each number $1,2,..., 100$ there exists at least one ticket with this number printed on it. Of course, there are tickets that have the same seat numbers. These $n$ spectators arrive one at a time. Each goes to the seat shown on his ticket and occupies it if it is still empty. If not, he just says “Oh” and moves to the seat with the next number. This is repeated until he finds an empty seat and occupies it, saying “Oh” once for each occupied seat passed over but not at any other time. Prove that all the spectators will be seated and that the total number of the exclamations “Oh” that have been made before all the spectators are seated does not depend on the order in which the n spectators arrive, although it does depend on the distribution of numbers on the tickets. (A Shen)

Let $n$ be a natural number. Determine the minimal number of equilateral triangles of side $1$ to cover the surface of an equilateral triangle of side $n +\frac{1}{2n}$.

A one-player game is played on a $m \times n$ table with $m \times n$ nuts. One of the nuts' sides is black, and the other side of them is white. In the beginning of the game, there is one nut in each cell of the table and all nuts have their white side upwards except one cell in one corner of the table which has the black side upwards. In each move, we should remove a nut which has its black side upwards from the table and reverse all nuts in adjacent cells (i.e. the cells which share a common side with the removed nut's cell). Find all pairs $(m,n)$ for which we can remove all nuts from the table.

Consider a convex polyhedron whose faces are triangles. Prove that it is possible to color its edges by either red or blue, in a way that the following property is satisfied: one can travel from any vertex to any other vertex while passing only along red edges, and can also do this while passing only along blue edges.

Let $A$ be the number of all points with integer coordinates in a three-dimensional coordinate system. We assume that nine arbitrary points in $A$ will be colored blue. Show that we can always find two blue dots so that the line segment between them contains at least one point from $A$.

There are two piles of stones with $1012$ stones each. Ann and Ben play a game. In every move, a player removes two stones from one of the piles and adds one to the other pile. Ann goes first. The first player to remove the last stone in one of the piles wins the game. Which player has a winning strategy and why?

In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $X$ is rational. [i]Proposed by Sammy Charney[/i] [b]Due to the problem having infinitely many solutions, all teams who inputted answers received points.[/b]

Through an informant, the police know the meeting place of a group of criminals. The identity of the different elements of the group is, however, unknown. Inspector Loureiro's task is to arrest the leader of the group. The inspector knows that the leader of the group is the shortest of the five members of the group, all of them of different heights, who will be present at the meeting. After the meeting, the bandits - as a precautionary measure - leave the building separately with an interval of $15$ minutes. As the inspector doesn't know which of them is the shortest, he decides to let the first two criminals out, and arrest the first of the following who is shorter than those who left until that moment. What is the probability that Inspector Loureiro will arrest the right person?

The magician puts out hardly a deck of $52$ cards and announces that $51$ of them will be thrown out of the table, and there will remain three of clubs. The viewer at each step says which count from the edge the card should be thrown out, and the magician chooses to count from the left or right edge, and ejects the corresponding card. At what initial positions of the three of clubs can the success of the focus be guaranteed?

A positive integer is called sparkly if it has exactly 9 digits, and for any n between 1 and 9 (inclusive), the nth digit is a positive multiple of n. How many positive integers are sparkly?

In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off). Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.