Given $2012$ distinct points $A_1,A_2,\dots,A_{2012}$ on the Cartesian plane. For any permutation $B_1,B_2,\dots,B_{2012}$ of $A_1,A_2,\dots,A_{2012}$ define the [i]shadow[/i] of a point $P$ as follows: [i]Point $P$ is rotated by $180^{\circ}$ around $B_1$ resulting $P_1$, point $P_1$ is rotated by $180^{\circ}$ around $B_2$ resulting $P_2$, ..., point $P_{2011}$ is rotated by $180^{\circ}$ around $B_{2012}$ resulting $P_{2012}$. Then, $P_{2012}$ is called the shadow of $P$ with respect to the permutation $B_1,B_2,\dots,B_{2012}$.[/i] Let $N$ be the number of different shadows of $P$ up to all permutations of $A_1,A_2,\dots,A_{2012}$. Determine the maximum value of $N$. [i]Proposer: Hendrata Dharmawan[/i]

$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others. $5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?

In the lower-left $3\times3$ square of an $8\times8$ chessboard there are nine pawns. Every pawn can jump horizontally or vertically over a neighboring pawn to the cell across it if that cell is free. Is it possible to arrange the nine pawns in the upperleft $3\times3$ square of the chessboard using finitely many such moves?

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Each BMT, every student chooses one of three focus rounds to take. Bob plans to attend BMT for the next $4$ years and wants to gure out what focus round to take each year. Given that he wants to take each focus round at least once, how many ways can he choose which round to take each year?

Jesse and Tjeerd are playing a game. Jesse has access to $n\ge 2$ stones. There are two boxes: in the black box there is room for half of the stones (rounded down) and in the white box there is room for half of the stones (rounded up). Jesse and Tjeerd take turns, with Jesse starting. Jesse grabs in his turn, always one new stone, writes a positive real number on the stone and places put him in one of the boxes that isn't full yet. Tjeerd sees all these numbers on the stones in the boxes and on his turn may move any stone from one box to the other box if it is not yet full, but he may also choose to do nothing. The game stops when both boxes are full. If then the total value of the stones in the black box is greater than the total value of the stones in the white box, Jesse wins; otherwise win Tjeerd. For every $n \ge 2$, determine who can definitely win (and give a corresponding winning strategy).

In the language of wolves has two letters $F$ and $P$, any finite sequence which forms a word. А word $Y$ is called 'subpart' of word $X$ if Y is obtained from X by deleting some letters (for example, the word $FFPF$ has 8 'subpart's: F, P, FF, FP, PF, FFP, FPF, FFF). Determine $n$ such that the $n$ is the greatest number of 'subpart's can have n-letter word language of wolves. F. Petrov, V. Volkov

Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k\minus{}1)\plus{}1$ sub-family such that in each sub-family, each pair of squares are disjoint.

Pasha placed numbers from 1 to 100 in the cells of the square $10\times 10$, each number exactly once. After that, Dima considered all sorts of squares, with the sides going along the grid lines, consisting of more than one cell, and painted in green the largest number in each such square (one number could be colored many times). Is it possible that all two-digit numbers are painted green?

[u]Round 1[/u] [b]p1.[/b] Let $ABCDEF$ be a regular hexagon. How many acute triangles have all their vertices among the vertices of $ABCDEF$? [b]p2.[/b] A rectangle has a diagonal of length $20$. If the width of the rectangle is doubled, the length of the diagonal becomes $22$. Given that the width of the original rectangle is $w$, compute $w^2$. [b]p3.[/b] The number $\overline{2022A20B22}$ is divisible by 99. What is $A + B$? [u]Round 2[/u] [b]p4.[/b] How many two-digit positive integers have digits that sum to at least $16$? [b]p5.[/b] For how many integers $k$ less than $10$ do there exist positive integers x and y such that $k =x^2 - xy + y^2$? [b]p6.[/b] Isosceles trapezoid $ABCD$ is inscribed in a circle of radius $2$ with $AB \parallel CD$, $AB = 2$, and one of the interior angles of the trapezoid equal to $110^o$. What is the degree measure of minor arc $CD$? [u]Round 3[/u] [b]p7.[/b] In rectangle $ALEX$, point $U$ lies on side $EX$ so that $\angle AUL = 90^o$. Suppose that $UE = 2$ and $UX = 12$. Compute the square of the area of $ALEX$. [b]p8.[/b] How many digits does $20^{22}$ have? [b]p9.[/b] Compute the units digit of $3 + 3^3 + 3^{3^3} + ... + 3^{3^{...{^3}}}$ , where the last term of the series has $2022$ $3$s. [u]Round 4[/u] [b]p10.[/b] Given that $\sqrt{x - 1} + \sqrt{x} = \sqrt{x + 1}$ for some real number $x$, the number $x^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Eric the Chicken Farmer arranges his $9$ chickens in a $3$-by-$3$ grid, with each chicken being exactly one meter away from its closest neighbors. At the sound of a whistle, each chicken simultaneously chooses one of its closest neighbors at random and moves $\frac12$ of a unit towards it. Given that the expected number of pairs of chickens that meet can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers, compute $p + q$. [b]p12.[/b] For a positive integer $n$, let $s(n)$ denote the sum of the digits of $n$ in base $10$. Find the greatest positive integer $n$ less than $2022$ such that $s(n) = s(n^2)$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949432p26408285]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings: (1) $\sum_{v\in V} f(v)=|E|$; (2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$. Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.

Dima invented a secret code in which every letter is replaced by a word no longer than $10$ letters. A code is called “good” if every encoded word can be decoded in only one way. Serjozha (with the help of a computer) checked that for Dima’s code, every possible word of at most $10000$ letters can be decoded in only one way. Does it follow that Dima’s code is good? (Note that Dima and Serjozha are Russian, so they use the Cyrillic alphabet, which has $ 33$ letters! A word is any sequence of letters.) (D Piontkovskiy, S Shalunov)

Consider a cake in the shape of a circle. It's been divided to some inequal parts by its radii. Arash and Bahram want to eat this cake. At the very first, Arash takes one of the parts. In the next steps, they consecutively pick up a piece adjacent to another piece formerly removed. Suppose that the cake has been divided to 5 parts. Prove that Arash can choose his pieces in such a way at least half of the cake is his.

Régis, Ed and Rafael are at the IMO. They are going to play a game in Bath, and there are $2^n$ houses in the city. Régis and Ed will team up against Rafael. The game operates as follows: First, Régis and Ed think on a strategy and then let Rafael know it. After this, Régis and Ed no longer communicate, and the game begins. Rafael decides on an order to visit the houses and then starts taking Régis to them in that order. At each house, except for the last one, Régis choose a number between $1$ and $n$ and places it in the house. In the last house, Rafael chooses a number from $1$ to $n$ and places it there. Afterwards, Ed sees all the houses and the numbers in them, and he must guess in which house Rafael placed the number. Ed is allowed $k$ guesses. What is the smallest $k$ for which there exists a strategy for Ed and Régis to ensure that Ed correctly guess the house where Rafael placed the number?

The main building of ETH Zurich is a rectangle divided into unit squares. Every side of a square is a wall, with certain walls having doors. The outer wall of the main building has no doors. A number of participants of the SMO have gathered in the main building lost. You can only move from one square to another through doors. We have indicates that there is a walkable path between every two squares of the main building. Cyril wants the participants to find each other again by having everyone on the same square leads. To do this, he can give them the following instructions via walkie-talkie: North, East, South or West. After each instruction, each participant simultaneously attempts a square in that direction to go. If there is no door in the corresponding wall, he remains standing. Show that Cyril can reach his goal after a finite number of directions, no matter which one square the participants at the beginning. [hide=original wording]Das Hauptgebäude der ETH Zürich ist ein in Einheitsquadrate unterteiltes Rechteck. Jede Seite eines Quadrates ist eine Wand, wobei gewisse Wände Türen haben. Die Aussenwand des Hauptgebäudes hat keine Türen. Eine Anzahl von Teilnehmern der SMO hat sich im Hauptgebäude verirrt. Sie können sich nur durch Türen von einem Quadrat zum anderen bewegen. Wir nehmen an, dass zwischen je zwei Quadraten des Hauptgebäudes ein begehbarer Weg existiert. Cyril möchte erreichen, dass sich die Teilnehmer wieder nden, indem er alle auf dasselbe Quadrat führt. Dazu kann er ihnen per Walkie-Talkie folgende Anweisungen geben: Nord, Ost, Süd oder West. Nach jeder Anweisung versucht jeder Teilnehmer gleichzeitig, ein Quadrat in diese Richtung zu gehen. Falls in der entsprechenden Wand keine Türe ist, bleibt er stehen. Zeige, dass Cyril sein Ziel nach endlich vielen Anweisungen erreichen kann, egal auf welchen Quadraten sich die Teilnehmer am Anfang benden. [/hide]

Let natural $n \ge 2$ be given. Let Laura be a student in a class of more than $n+2$ students, all of which participated in an olympiad and solved some problems. Additionally, it is known that: [list] [*] for every pair of students there is exactly one problem that was solved by both students; [*] for every pair of problems there is exactly one student who solved both of them; [*] one specific problem was solved by Laura and exactly $n$ other students. [/list] Determine the number of students in Laura's class.

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]D1 / Z1.[/b] What is $1 + 2 \cdot 3$? [b]D2.[/b] What is the average of the first $9$ positive integers? [b]D3 / Z2.[/b] A square of side length $2$ is cut into $4$ congruent squares. What is the perimeter of one of the $4$ squares? [b]D4.[/b] Find the ratio of a circle’s circumference squared to the area of the circle. [b]D5 / Z3.[/b] $6$ people split a bag of cookies such that they each get $21$ cookies. Kyle comes and demands his share of cookies. If the $7$ people then re-split the cookies equally, how many cookies does Kyle get? [u]Set 2[/u] [b]D6.[/b] How many prime numbers are perfect squares? [b]D7.[/b] Josh has an unfair $4$-sided die numbered $1$ through $4$. The probability it lands on an even number is twice the probability it lands on an odd number. What is the probability it lands on either $1$ or $3$? [b]D8.[/b] If Alice consumes $1000$ calories every day and burns $500$ every night, how many days will it take for her to first reach a net gain of $5000$ calories? [b]D9 / Z4.[/b] Blobby flips $4$ coins. What is the probability he sees at least one heads and one tails? [b]D10.[/b] Lillian has $n$ jars and $48$ marbles. If George steals one jar from Lillian, she can fill each jar with $8$ marbles. If George steals $3$ jars, Lillian can fill each jar to maximum capacity. How many marbles can each jar fill? [u]Set 3[/u] [b]D11 / Z6.[/b] How many perfect squares less than $100$ are odd? [b]D12.[/b] Jash and Nash wash cars for cash. Jash gets $\$6$ for each car, while Nash gets $\$11$ per car. If Nash has earned $\$1$ more than Jash, what is the least amount of money that Nash could have earned? [b]D13 / Z5.[/b] The product of $10$ consecutive positive integers ends in $3$ zeros. What is the minimum possible value of the smallest of the $10$ integers? [b]D14 / Z7.[/b] Guuce continually rolls a fair $6$-sided dice until he rolls a $1$ or a $6$. He wins if he rolls a $6$, and loses if he rolls a $1$. What is the probability that Guuce wins? [b]D15 / Z8.[/b] The perimeter and area of a square with integer side lengths are both three digit integers. How many possible values are there for the side length of the square? PS. You should use hide for answers. D.16-30/Z.9-14, 17, 26-30 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider all $26^{26}$ words of length 26 in the Latin alphabet. Define the $\emph{weight}$ of a word as $1/(k+1)$, where $k$ is the number of letters not used in this word. Prove that the sum of the weights of all words is $3^{75}$. Proposed by Fedor Petrov, St. Petersburg State University

Alice is wandering in the country of Wanderland. Wanderland consists of a finite number of cities, some of which are connected by two-way trains, such that Wanderland is connected: given any two cities, there is always a way to get from one to the other through a series of train rides. Alice starts at Riverbank City and wants to end up at Conscious City. Every day, she picks a train going out of the city she is in uniformly at random among all of the trains, and then boards that train to the city it leads to. Show that the expected number of days it takes for her to reach Conscious City is finite.

Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied: (1) only digits $1$ and $2$ are used; (2) each number consists of exactly $n$ digits; (3) different numbers are assigned to different vertices; (4) the numbers assigned to two neighboring vertices differ at exactly one digit.

The numbers $1,2,3,\dots,1000$ are written on the board. Patya and Vassya are playing a game. They take turn alternatively erasing a number from the board. Patya begins. If after a turn all numbers (maybe one) on the board be divisible by a natural number greater than $1$ the player who last played loses. If after some number of steps the only remaining number on the board be $1$ then they call it a draw. Determine the result of the game if they both play their best.

In a company of $6$ sousliks each souslik has $4$ friends. Is it always possible to divide this company into two groups of $3$ sousliks such that in both groups all sousliks are friends?