Let $n>2$ be a given positive integer. There are $n$ guests at Georg's bachelor party and each guest is friends with at least one other guest. Georg organizes a party game among the guests. Each guest receives a jug of water such that there are no two guests with the same amount of water in their jugs. All guests now proceed simultaneously as follows. Every guest takes one cup for each of his friends at the party and distributes all the water from his jug evenly in the cups. He then passes a cup to each of his friends. Each guest having received a cup of water from each of his friends pours the water he has received into his jug. What is the smallest possible number of guests that do not have the same amount of water as they started with?

$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.

[u]Round 1 [/u] [b]p1. [/b]Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the 12th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p2.[/b] Show that in the sequence $10017$, $100117$, $1001117$, $...$ all numbers are divisible by $53$. [b]p3.[/b] Harry and Draco have three wands: a bamboo wand, a willow wand, and a cherry wand, all of the same length. They must perform a spell wherein they take turns picking a wand and breaking it into three parts – first Harry, then Draco, then Harry again. But in order for the spell to work, Harry has to make sure it is possible to form three triangles out of the pieces of the wands, where each triangle has a piece from each wand. How should he break the wands to ensure the success of the spell? [b]p4.[/b] A $2\times 2\times 2$ cube has $4$ equal squares on each face. The squares that share a side are called neighbors (thus, each square has $4$ neighbors – see picture). Is it possible to write an integer in each square in such a way that the sum of each number with its $4$ neighbors is equal to $13$? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/8/4/0f7457f40be40398dee806d125ba26780f9d3a.png[/img] [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2 [/u] [b]p6.[/b] A red square is placed on a table. $2010$ white squares, each the same size as the red square, are then placed on the table in such a way that the red square is fully covered and the sides of every white square are parallel to the sides of the red square. Is it always possible to remove one of the white squares so the red square remains completely covered? [b]p7.[/b] A computer starts with a given positive integer to which it randomly adds either $54$ or $77$ every second and prints the resulting sum after each addition. For example, if the computer is given the number $1$, then a possible output could be: $1$, $55$, $109$, $186$, $…$ Show that after finitely many seconds the computer will print a number whose last two digits are the same. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Compute $$\sqrt{(\sqrt{63} +\sqrt{112} +\sqrt{175})(-\sqrt{63} +\sqrt{112} +\sqrt{175})(\sqrt{63}-\sqrt{112} +\sqrt{175})(\sqrt{63} +\sqrt{112} -\sqrt{175})}$$ [b]p2.[/b] Consider the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many distinct $3$-element subsets are there such that the sum of the elements in each subset is divisible by $3$? [b]p3.[/b] Let $a^2$ and $b^2$ be two integers. Consider the triangle with one vertex at the origin, and the other two at the intersections of the circle $x^2 + y^2 = a^2 + b^2$ with the graph $ay = b|x|$. If the area of the triangle is numerically equal to the radius of the circle, what is this area? [b]p4.[/b] Suppose $f(x) = x^3 + x - 1$ has roots $a$, $b$ and $c$. What is $$\frac{a^3}{1-a}+\frac{b^3}{1-b}+\frac{c^3}{1-c} ?$$ [b]p5.[/b] Lisa has a $2D$ rectangular box that is $48$ units long and $126$ units wide. She shines a laser beam into the box through one of the corners such that the beam is at a $45^o$ angle with respect to the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the four corners of the box. [b]p6.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people total? Express your answer in the form $a^b + c$, where $a$, $b$, and $c$ are integers, and $a$ is prime. [b]p7.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ...\log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$. [b]p8.[/b] A prison, housing exactly four hundred prisoners in four hundred cells numbered $1$-$400$, has a really messed-up warden. One night, when all the prisoners are asleep and all of their doors are locked, the warden toggles the locks on all of their doors (that is, if the door is locked, he unlocks the door, and if the door is unlocked, he locks it again), starting at door $1$ and ending at door $400$. The warden then toggles the lock on every other door starting at door $2$ ($2$, $4$, $6$, etc). After he has toggled the lock on every other door, the warden then toggles every third door (doors $3$, $6$, $9$, etc.), then every fourth door, etc., finishing by toggling every $400$th door (consisting of only the $400$th door). He then collapses in exhaustion. Compute the number of prisoners who go free (that is, the number of unlocked doors) when they wake up the next morning. [b]p9.[/b] Let $A$ and $B$ be fixed points on a $2$-dimensional plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ on the same plane and finds that the distance from itself to $B$ always decreases at any time during this walk. Compute the area of the locus of points where point $C$ could possibly be located. [b]p10.[/b] A robot starts in the bottom left corner of a $4 \times 4$ grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step? [b]p11.[/b] Assuming real values for $p$, $q$, $r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Find $q$. [b]p12.[/b] A cube is inscribed in a right circular cone such that one face of the cube lies on the base of the cone. If the ratio of the height of the cone to the radius of the cone is $2 : 1$, what fraction of the cone's volume does the cube take up? Express your answer in simplest radical form. [b]p13.[/b] Let $$y =\dfrac{1}{1 +\dfrac{1}{9 +\dfrac{1}{5 +\dfrac{1}{9 +\dfrac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$, where $b$ is not divisible by the square of any prime, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$. [b]p14.[/b] Alice wants to paint each face of an octahedron either red or blue. She can paint any number of faces a particular color, including zero. Compute the number of ways in which she can do this. Two ways of painting the octahedron are considered the same if you can rotate the octahedron to get from one to the other. [b]p15.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5,$$ where $n$ is an integer less than $170$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are eight people, each with their own horse. The horses are arbitrarily arranged in a line from left to right, while the people are lined up in random order to the left of all the horses. One at a time, each person moves rightwards in an attempt to reach their horse. If they encounter a mounted horse on their way to their horse, the mounted horse shouts angrily at the person, who then scurries home immediately. Otherwise, they get to their horse safely and mount it. The expected number of people who have scurried home after all eight people have attempted to reach their horse can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.

$100$ points at a circle with radius $1$ $cm$. Show that, we find an another point such that, this point's distance to other $100$ points is greater than $100$ $cm$.

$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?

Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd.

Consider an $n\times n$ unit-square board. The main diagonal of the board is the $n$ unit squares along the diagonal from the top left to the bottom right. We have an unlimited supply of tiles of this form: [asy] size(1.5cm); draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0)); [/asy] The tiles may be rotated. We wish to place tiles on the board such that each tile covers exactly three unit squares, the tiles do not overlap, no unit square on the main diagonal is covered, and all other unit squares are covered exactly once. For which $n\geq 2$ is this possible? [i]Proposed by Daniel Kohen[/i]

Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number $1221$ is a palindrome, but the numbers $1231, 1212$ and $1010$ are not). Which palindromes among the numbers from $10,000$ to $999,999$ have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?

Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the permutation, the number of numbers less than $k$ that follow $k$ is even. For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$, then find the product $ab$.

Find the largest positive integer $N$ such that the number of integers In the set ${1,2,3,...,N}$ which are divisible by $3$ is equal to the number of integers which are divisible by $5$ or $7$ (or both),

Consider the string of length $ 6$ composed of three characters $ a$, $ b$, $ c$. For each string, if two $ a$s are next to each other, or two $ b$s are next to each other, then replace $ aa$ by $ b$, and replace $ bb$ by $ a$. Also, if $ a$ and $ b$ are next to each other, or two $ c$s are next to each other, remove all two of them (i.e. delete $ ab$, $ ba$, $ cc$). Determine the number of strings that can be reduced to $ c$, the string of length 1, by the reducing processes mentioned above.

Determine the largest integer $N$, for which there exists a $6\times N$ table $T$ that has the following properties: $*$ Every column contains the numbers $1,2,\ldots,6$ in some ordering. $*$ For any two columns $i\ne j$, there exists a row $r$ such that $T(r,i)= T(r,j)$. $*$ For any two columns $i\ne j$, there exists a row $s$ such that $T(s,i)\ne T(s,j)$. (Proposed by Gerhard Woeginger, Austria)

The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the "right combination" is not known)?

Morteza and Amir Reza play the following game. First each of them independently roll a dice $100$ times in a row to construct a $100$-digit number with digits $1,2,3,4,5,6$ then they simultaneously shout a number from $1$ to $100$ and write down the corresponding digit to the number other person shouted in their $100$ digit number. If both of the players write down $6$ they both win otherwise they both loose. Do they have a strategy with wining chance more than $\frac{1}{36}$? [i]Proposed by Morteza Saghafian[/i]

Members of the State Duma formed factions in such a way that for any two fractions $A $ and $B$ (not necessarily different), $\overline{A \cup B}$ is also faction ($\overline{C}$ denotes the set of all members of the Duma, not including in $C$). Prove that for any two factions $A$ and $B$, $A \cup B$ is also a faction.

We are given an angle $0^o < \phi \le 180^o$ and a circular disc. An ant begins its journey from an interior point of the disc, travelling in a straight line in a certain direction. When it reaches the edge of the disc, it does the following: it turns clockwise by the angle $\phi $, and if its new direction does not point towards the interior of the disc, it turns by the angle $\phi $ again, and repeats this until it faces the interior. Then it continues its journey in this new direction and turns as before every time when it reaches the edge. For what values of $\phi $ is it true that for any starting point and initial direction the ant eventually returns to its starting position?

Points $P_1,P_2,\ldots,P_n,Q$ are given in space $(n\ge4)$, no four of which are in a plane. Prove that if for any three distinct points $P_\alpha,P_\beta,P_\gamma$ there is a point $P_\delta$ such that the tetrahedron $P_\alpha P_\beta P_\gamma P_\delta$ contains the point $Q$, then $n$ is an even number.

7. There are 100 chess bishops on white squares of a $100 \times 100$ chess board. Some of them are white and some of them are black. They can move in any order and capture the bishops of opposing color. What number of moves is sufficient for sure to retain only one bishop on the chess board?

A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$

The PMO Magician has a special party game. There are $n$ chairs, labelled $1$ to $n$. There are $n$ sheets of paper, labelled $1$ to $n$. [list] [*] On each chair, she attaches exactly one sheet whose number does not match the number on the chair. [*] She then asks $n$ party guests to sit on the chairs so that each chair has exactly one occupant. [*] Whenever she claps her hands, each guest looks at the number on the sheet attached to their current chair, and moves to the chair labelled with that number. [/list] Show that if $1 < m \leq n$, where $m$ is not a prime power, it is always possible for the PMO Magician to choose which sheet to attach to each chair so that everyone returns to their original seats after exactly $m$ claps.

Ten square cardboards of $3$ centimeters on a side are cut by a line, as indicated in the figure. After the cuts, there are $20$ pieces: $10$ triangles and $10$ trapezoids. Assemble a square that uses all $20$ pieces without overlaps or gaps. [img]https://cdn.artofproblemsolving.com/attachments/7/9/ec2242cca617305b02eef7a5409e6a6b482d66.gif[/img]

Let a city be in the form of a square grid which has $n \times n$ cells, each of which contain a skyscraper . At first the $m$ skyscrapers burn, but the fire spreads: everyone skyscraper that has at least two burning neighboring houses (by neighboring houses we mean only houses that border the house along a street, not just at a corner) immediately gets fire. Prove that when in the end the whole city burns down, of must have been $m \ge n$. [hide=original wording in German] Eine Stadt habe die Form eines quadratischen Gitters, welches n × n Zellen habe, von denen jede ein Hochhaus enthalte. Anfangs brennen m der Hochh¨auser, doch der Brand pflanzt sich fort: Jedes Hochhaus, das mindestens zwei brennende Nachbarh¨auser hat (unter Nachbarh¨ausern verstehen wir dabei nur H¨auser, die entlang einer Straße an das Haus angrenzen, nicht nur an einer Ecke), f¨angt sofort Feuer. Man beweise: Wenn am Ende die gesamte Stadt abgebrannt ist,muss m ≥ n gewesen sein.[/hide]

If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting.