There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered $i, j, k$, in order. 1) If $i>j>k$, then the ball $j$ is painted in red; 2) If $i<j<k$, then the ball $j$ is painted in yellow; 3) If $i<j, k<j$, then the ball $j$ is painted in blue; 4) If $i>j, k>j$, then the ball $j$ is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods. Find out the number of all distinct painting methods.

The number $1234567890$ is written on the blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In one move, a player erases the number which is written on the blackboard, say, $m$, subtracts from $m$ any positive integer not exceeding the sum of the digits of $m$ and writes the obtained result instead of $m$. The first player who reduces the number written on the blackboard to $0$ wins. Determine which of the players has the winning strategy if the player $A$ makes the first move.

In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1,C_2,C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1,C_3$ white and $C_2$ red. Find the maximum value $T$ can attain. [i]Proposed by Merlijn Staps, The Netherlands[/i]

Six young people - Antonio, Bernardo, Carlos, Diego, Eduardo, and Francisco, attended a meeting in vests of different colors. After the meeting, they decided to exchange the vests as souvenir. $1)$. Each of them came out of the meeting room, wearing a vest with color different from the one with which they went into the meeting room. $2)$. The vest with which Antonio came out of the meeting room was belong to the young man who came out with Bernardo's vest. $3)$. The owner of the vest with which Carlos came out of the meeting room, came out with the vest that was belong to the young man who came out with Diego's vest. $4)$. The one who came out of the meeting room with Eduardo's vest was not the owner of the vest with which Francisco came out. Determine who came out of the meeting room with Antonio's vest, and who owns the vest with which Antonio came out. [hide=original wording]Seis jovenes que asistieron a una reunion vistiendo chalecos de distintos colores, decidieron intercambiarlos y salieron vistiendo todos de color diferente a aquel con que llegaron. El chaleco con que salio Antonio perteneca al joven que salio con el chaleco de Bernardo. El dueno del chaleco con que salio Carlos, salio con el chaleco que perteneca al joven que se llevo el de Diego. Quien se llevo el chaleco de Eduardo no era el dueno del que se llevo Francisco. Determine quien salio con el chaleco de Antonio, y quien es el dueno del chaleco que se llevo Antonio.[/hide]

Let $A$ be a point in the plane and $3$ lines which pass through this point divide the plane in $6$ regions. In each region there are $5$ points. We know that no three of the $30$ points existing in these regions are collinear. Prove that there exist at least $1000$ triangles whose vertices are points of those regions such that $A$ lies either in the interior or on the side of the triangle.

Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.) [i](6 points)[/i]

On a circle the numbers from 1 to 6 are written in order, as depicted in the picture. In each move, Lior picks a number $a$ on the circle whose neighbors are $b$ and $c$ and replaces it by the number $\frac{bc}{a}$. Can Lior reach a state in which the product of the numbers on the circle is greater than $10^{100}$ in [b]a)[/b] at most 100 moves [b]b)[/b] at most 110 moves

On circle there are $2009$ positive integers which sum is $7036$. Show that it is possible to find two pairs of neighboring numbers such that sum of both pairs is greater or equal to $8$

Let $ S$ be the set of all odd positive integers less than $ 30m$ which are not multiples of $ 5$, where $ m$ is a given positive integer. Find the smallest positive integer $ k$ such that each $ k$-element subset of $ S$ contains two distinct numbers, one of which divides the other.

There is a city with many houses, where the houses are connected by some two-way roads. It is known that for any two houses $A,B$, there is exactly one house $C$ such that both $A,B$ are connected to $C$. Show that for any two houses not connected directly by a road, they have the same number of roads adjacent to them. [i]ST[/i]

In a finite, simple, connected graph $G$ we play the following game: initially we color all the vertices with a different color. In each step we choose a vertex randomly (with uniform distribution), and then choose one of its neighbors randomly (also with uniform distribution), and color it to the the same color as the originally chosen vertex (if the two chosen vertices already have the same color, we do nothing). The game ends when all the vertices have the same color. Knowing graph $G$ find the probability for each vertex that the game ends with all vertices having the same color as the chosen vertex.

Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.

You are given a checkered square, the side of which is $n – 1$ long and contains $n \geq 10$ nodes. A non-return path is a path along edges, the intersection of which with any horizontal or vertical line is a segment, point or empty set, and which does not pass along any edge more than once. What is the smallest number of non-return paths that can cover all the edges? (An edge is a unit segment between adjacent nodes.)

In the game of Yahtzee , players have to achieve various combinations of values with $5$ dice. In a round, a player can roll the dice three times. At the second and third rolls, he can choose which dice to re-roll and which to keep. What is the probability that a player achieves at least four $6$’s in a round, given that he plays with the optimal strategy to maximise this probability? Writing the answer as $p/q$ where $p$ and $q$ are coprime, you should submit the sum of all prime factors of $p$, counted with multiplicity. So for example if you obtained $\frac{p}{q} = \frac{3^4 \cdot 11}{ 2^5 \cdot 5}$ then the submitted answer should be $4 \cdot 3 + 11 = 23$.

Let $n$ be a natural number. A tiling of a $2n \times 2n$ board is a placing of $2n^2$ dominos (of size $2 \times 1$ or $1 \times 2$) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two [i]sepearate tilings[/i] of a $2n \times 2n$ board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours. Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that $n$ is divisible by $3$ [i] Proposed by Tejaswi Navilarekallu [/i]

Let $n \geq 2$ be a positive integer. Determine the number of $n$-tuples $(x_1, x_2, \ldots, x_n)$ such that $x_k \in \{0, 1, 2\}$ for $1 \leq k \leq n$ and $\sum_{k = 1}^n x_k - \prod_{k = 1}^n x_k$ is divisible by $3$.

$A$ is a set satisfying the following the condition. Show that $2001+\sqrt{2001}$ is an element of $A$. [b]Condition[/b] (1) $1 \in A$ (2) If $x \in A$, then $x^2 \in A$. (3) If $(x-3)^2 \in A$, then $x \in A$.

$3$ green, $4$ yellow, and $5$ red balls are placed in a bag. (Large piles of balls of each colour are outside the bag.) Two balls of different colours are selected at random, and replaced by two balls of the third colour. If, at some point, there are $5$ green balls left in the bag, and there are at least as many yellow balls as red balls left in the bag, how many balls of each colour are left in the bag? Write your answer in the form $(g,y, r)$, where $g$ is the number of green balls and so on.

There are 21 competitors with distinct skill levels numbered 1, 2,..., 21. They participate in a ping-pong tournament as follows. First, a random competitor is chosen to be "active", while the rest are "inactive." Every round, a random inactive competitor is chosen to play against the current active one. The player with the higher skill will win and become (or remain) active, while the loser will be eliminated from the tournament. The tournament lasts for 20 rounds, after which there will only be one player remaining. Alice is the competitor with skill 11. What is the expected number of games that she will get to play?

Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?

Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.

Worms grow at the rate of $1$ metre per hour. When they reach their maximal length of $1$ metre, they stop growing. A full-grown worm may be dissected into two not necessarily equal parts. Each new worm grows at the rate of $1$ metre per hour. Starting with $1$ full-grown worm, can one obtain $10$ full-grown worms in less than $1$ hour?

[b]p1.[/b] Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation $x^{17} - 17x + 17 = 0$. [b]p2.[/b] Four identical spherical cows, each of radius $17$ meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow? [b]p3.[/b] If $a_n$ is the last digit of $\sum^{n}_{i=1} i$, what would the value of $\sum^{1000}_{i=1}a_i$ be? [b]p4.[/b] If there are $15$ teams to play in a tournament, $2$ teams per game, in how many ways can the tournament be organized if each team is to participate in exactly $5$ games against dierent opponents? [b]p5.[/b] For $n = 20$ and $k = 6$, calculate $$2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}$$ where ${n \choose k}$ is the number of ways to choose $k$ things from a set of $n$. [b]p6.[/b] Given a function $f(x) = ax^2 + b$, with a$, b$ real numbers such that $$f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}$$ , find $b^a$. [b]p7.[/b] Simplify the following fraction $$\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}$$ [b]p8.[/b] Simplify the following expression $$\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}$$ [b]p9.[/b] Suppose that $p(x)$ is a polynomial of degree $100$ such that $p(k) = k2^{k-1}$ , $k =1, 2, 3 ,... , 100$. What is the value of $p(101)$ ? [b]p10. [/b] Find all $17$ real solutions $(w, x, y, z)$ to the following system of equalities: $$ 2w + w^2x = x$$ $$ 2x + x^2y=y $$ $$ 2y + y^2z=z $$ $$ -2z+z^2w=w $$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

Seven cards numbered $1$ through $7$ lay stacked in a pile in ascending order from top to bottom ($1$ on top, $7$ on bottom). A shuffle involves picking a random card [i]of the six not currently on top[/i], and putting it on top. The relative order of all the other cards remains unchanged. Find the probability that, after $10$ shuffles, $6$ is higher in the pile than $3$.