[i]Nocycleland[/i] is a country with $500$ cities and $2013$ two-way roads, each one of them connecting two cities. A city $A$ [i]neighbors[/i] $B$ if there is one road that connects them, and a city $A$ [i]quasi-neighbors[/i] $B$ if there is a city $C$ such that $A$ neighbors $C$ and $C$ neighbors $B$. It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities $A$, $B$, $C$ and $D$ such that $A$ neighbors $B$, $B$ neighbors $C$, $C$ neighbors $D$, and $D$ neighbors $A$. Show that there is at least one city that quasi-neighbors at least $57$ other cities.

We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$

Prove that the number $ (\underbrace{9999 \dots 99}_{2005}) ^{2009}$ can be obtained by erasing some digits of $ (\underbrace{9999 \dots 99}_{2008}) ^{2009}$ (both in decimal representation). [i]Yudi Satria, Jakarta[/i]

$N^3$ unit cubes are made into beads by drilling a hole through them along a diagonal, put on a string and binded. Thus the cubes can move freely in space as long as the vertices of two neighboring cubes (including the first and last one) are touching. For which $N$ is it possible to build a cube of edge $N$ using these cubes?

[u]Set 6[/u] [b]p16.[/b] Let $x$ and $y$ be non-negative integers. We say point $(x, y)$ is square if $x^2 + y$ is a perfect square. Find the sum of the coordinates of all distinct square points which also satisfy $x^2 + y \le 64$. [b]p17.[/b] Two integers $a$ and $b$ are randomly chosen from the set $\{1, 2, 13, 17, 19, 87, 115, 121\}$, with $a > b$. What is the expected value of the number of factors of $ab$? [b]p18.[/b] Marnie the Magical Cello is jumping on nonnegative integers on number line. She starts at $0$ and jumps following two specific rules. For each jump she can either jump forward by $1$ or jump to the next multiple of $4$ (the next multiple must be strictly greater than the number she is currently on). How many ways are there for her to jump to $2022$? (Two ways are considered distinct only if the sequence of numbers she lands on is different.) PS. You should use hide for answers.

The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with $50\%$ probability and scissors with $50\%$ probability. If you play optimally against the computer, find the probability that after $8$ games you have won at least $4$. [i]In the game of rock-paper-scissors, two players each choose one of rock, paper, or scissors to play. Rock beats scissors, scissors beats paper, and paper beats rock. If the players play the same thing, the match is considered a draw.[/i]

[u]Round 1[/u] [b]p1.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p2.[/b] A positive number is placed on each of the $10$ circles in this picture. It turns out that for each of the nine little equilateral triangles, the number on one of its corners is the sum of the numbers on the other two corners. Is it possible that all $10$ numbers are different? [img]https://cdn.artofproblemsolving.com/attachments/b/f/c501362211d1c2a577e718d2b1ed1f1eb77af1.png[/img] [b]p3.[/b] Pablo and Nina take turns entering integers into the cells of a $3 \times 3$ table. Pablo goes first. The person who fills the last empty cell in a row must make the numbers in that row add to $0$. Can Nina ensure at least two of the columns have a negative sum, no matter what Pablo does? [b]p4. [/b]All possible simplified fractions greater than $0$ and less than $1$ with denominators less than or equal to $100$ are written in a row with a space before each number (including the first). Zeke and Qing play a game, taking turns choosing a blank space and writing a “$+$” or “$-$” sign in it. Zeke goes first. After all the spaces have been filled, Zeke wins if the value of the resulting expression is an integer. Can Zeke win no matter what Qing does? [img]https://cdn.artofproblemsolving.com/attachments/3/6/15484835686fbc2aa092e8afc6f11cd1d1fb88.png[/img] [b]p5.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/0/c/d827cf26c8eaabfd5b0deb92612a6e6ebffb47.png[/img] [u]Round 2[/u] [b]p6.[/b] Prove that among any $3^{2022}$ integers, it is possible to find exactly $3^{2021}$ of them whose sum is divisible by $3^{2021}$. [b]p7.[/b] Given a list of three numbers, a zap consists of picking two of the numbers and decreasing each of them by their average. For example, if the list is $(5, 7, 10)$ and you zap $5$ and $10$, whose average is $7.5$, the new list is $(-2.5, 7, 2.5)$. Is it possible to start with the list $(3, 1, 4)$ and, through some sequence of zaps, end with a list in which the sum of the three numbers is $0$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the smallest positive integer $k{}$ satisfying the following condition: For any configuration of chess queens on a $100 \times 100$ chequered board, the queens can be coloured one of $k$ colours so that no two queens of the same colour attack each other. [i]Russia, Sergei Avgustinovich and Dmitry Khramtsov[/i]

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Each square of a $7 \times 8$ board is painted black or white, in such a way that each $3 \times 3$ subboard has at least two black squares that are neighboring. What is the least number of black squares that can be on the entire board? Clarification: Two squares are [i]neighbors [/i] if they have a common side.

There are $n$ cities in a country, where $n>1$. There are railroads connecting some of the cities so that you can travel between any two cities through a series of railroads (railroads run in both direction.) In addition, in this country, it is impossible to travel from a city, through a series of distinct cities, and return back to the original city. We define the [b]degree[/b] of a city as the number of cities directly connected to it by a single segment of railroad. For a city $A$ that is directly connected to $x$ cities, with $y$ of those cities having a smaller degree than city $A$, the [b]significance[/b] of city $A$ is defined as $\frac{y}{x}$. Find the smallest positive real number $t$ so that, for any $n>1$, the sum of the significance of all cities is less than $tn$, no matter how the railroads are paved. [i]Proposed by houkai[/i]

[b]p1.[/b] If $a \diamond b = ab - a + b$, find $(3 \diamond 4) \diamond 5$ [b]p2.[/b] If $5$ chickens lay $5$ eggs in $5$ days, how many chickens are needed to lay $10$ eggs in $10$ days? [b]p3.[/b] As Alissa left her house to go to work one hour away, she noticed that her odometer read $16261$ miles. This number is a "special" number for Alissa because it is a palindrome and it contains exactly $1$ prime digit. When she got home that evening, it had changed to the next greatest "special" number. What was Alissa's average speed, in miles per hour, during her two hour trip? [b]p4.[/b] How many $1$ in by $3$ in by $8$ in blocks can be placed in a $4$ in by $4$ in by $9$ in box? [b]p5.[/b] Apple loves eating bananas, but she prefers unripe ones. There are $12$ bananas in each bunch sold. Given any bunch, if there is a $\frac13$ probability that there are $4$ ripe bananas, a $\frac16$ probability that there are $6$ ripe bananas, and a $\frac12$ probability that there are $10$ ripe bananas, what is the expected number of unripe bananas in $12$ bunches of bananas? [b]p6.[/b] The sum of the digits of a $3$-digit number $n$ is equal to the same number without the hundreds digit. What is the tens digit of $n$? [b]p7.[/b] How many ordered pairs of positive integers $(a, b)$ satisfy $a \le 20$, $b \le 20$, $ab > 15$? [b]p8.[/b] Let $z(n)$ represent the number of trailing zeroes of $n!$. What is $z(z(6!))?$ (Note: $n! = n\cdot (n-1) \cdot\cdot\cdot 2 \cdot 1$) [b]p9.[/b] On the Cartesian plane, points $A = (-1, 3)$, $B = (1, 8)$, and $C = (0, 10)$ are marked. $\vartriangle ABC$ is reflected over the line $y = 2x + 3$ to obtain $\vartriangle A'B'C'$. The sum of the $x$-coordinates of the vertices of $\vartriangle A'B'C'$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Compute $a + b$. [b]p10.[/b] How many ways can Bill pick three distinct points from the figure so that the points form a non-degenerate triangle? [img]https://cdn.artofproblemsolving.com/attachments/6/a/8b06f70d474a071b75556823f70a2535317944.png[/img] [b]p11.[/b] Say piece $A$ is attacking piece $B$ if the piece $B$ is on a square that piece $A$ can move to. How many ways are there to place a king and a rook on an $8\times 8$ chessboard such that the rook isn't attacking the king, and the king isn't attacking the rook? Consider rotations of the board to be indistinguishable. (Note: rooks move horizontally or vertically by any number of squares, while kings move $1$ square adjacent horizontally, vertically, or diagonally). [b]p12.[/b] Let the remainder when $P(x) = x^{2020} - x^{2017} - 1$ is divided by $S(x) = x^3 - 7$ be the polynomial $R(x) = ax^2 + bx + c$ for integers $a$, $b$, $c$. Find the remainder when $R(1)$ is divided by $1000$. [b]p13.[/b] Let $S(x) = \left \lfloor \frac{2020}{x} \right\rfloor + \left \lfloor \frac{2020}{x + 1} \right\rfloor$. Find the number of distinct values $S(x)$ achieves for integers $x$ in the interval $[1, 2020]$. [b]p14.[/b] Triangle $\vartriangle ABC$ is inscribed in a circle with center $O$ and has sides $AB = 24$, $BC = 25$, $CA = 26$. Let $M$ be the midpoint of $\overline{AB}$. Points $K$ and $L$ are chosen on sides $\overline{BC}$ and $\overline{CA}$, respectively such that $BK < KC$ and $CL < LA$. Given that $OM = OL = OK$, the area of triangle $\vartriangle MLK$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p15.[/b] Euler's totient function, $\phi (n)$, is defined as the number of positive integers less than $n$ that are relatively prime to $n$. Let $S(n)$ be the set of composite divisors of $n$. Evaluate $$\sum^{50}_{k=1}\left( k - \sum_{d\in S(k)} \phi (d) \right)$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]

The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$

In the country of [i]Sugarland[/i], there are $13$ students in the IMO team selection camp. $6$ team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these $6$ tests and starting from the first test, the person with the highest score between the remaining students will become a member of the team.The committee is having a session to choose the permutation. Is it possible that all $13$ students have a chance of being a team member? [i]Proposed by Morteza Saghafian[/i]

Petya chooses $100$ pairwise distinct positive numbers less than $1$ and arranges them in a circle. In one operation, he may take three consecutive numbers \( a, b, c \) (in this order) and replace \( b \) with \( a - b + c \). What is the greatest value of \( k \) such that Petya could initially choose the numbers and perform several operations so that \( k \) of the resulting numbers are integers? \\

You are given $20$ weights such that any object of integer weight $m$, $1 \le m \le1997$, can be balanced by placing it on one pan of a balance and a subset of the weights on the other pan. What is the minimal value of largest of the $20$ weights if the weights are (a) all integers; (b) not necessarily integers? (M Rasin)

On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region. (P Kozhevnikov)

A rectangular array of positive integers has $4$ rows. The sum of the entries in each column is $20$. Within each row, all entries are distinct. What is the maximum possible number of columns?

In Tlaxcala, there is a transportation system that works through buses that travel from one city to another in one direction . A set $S$ of cities is said [i]beautiful[/i] if it contains at least three different cities and from each city $A$ in $S$ at least two buses depart, each one goes directly to a different city in $S$ and none of them is $A$ (if there is a direct bus from $A$ to a city $B$ in $S$, there is not necessarily a direct bus from $B$ to $A$). Show that if there exists a beautiful set of cities $S$, then there exists a beautiful $T$ subset of $S$, such that for any two cities in $T$, you can get from one to another by taking buses that only pass through cities in $T$. Note: A bus goes directly from one city to another if it does not pass through any other city.

Let n be the natural number ($n\ge 2$). All natural numbers from $1$ up to $n$ ,inclusive, are written on the board in some order: $a_1$, $a_2$ , $...$ , $a_n$. Determine all natural numbers $n$ ($n\ge 2$), for which the product $$P = (1 + a_1) \cdot (2 + a_2) \cdot ... \cdot (n + a_n)$$ is an even number, whatever the arrangement of the numbers written on the board.

Given is an integer $k\geq 2$. Determine the smallest positive integer $n$, such that, among any $n$ points in the plane, there exist $k$ points among them, such that all distances between them are less than or equal to $2$, or all distances are strictly greater than $1$.

[b]p1.[/b] Frankie the frog likes to hop. On his first hop, he hops $1$ meter. On each successive hop, he hops twice as far as he did on the previous hop. For example, on his second hop, he hops $2$ meters, and on his third hop, he hops $4$ meters. How many meters, in total, has he travelled after $6$ hops? [b]p2.[/b] Anton flips $5$ fair coins. The probability that he gets an odd number of heads can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p3.[/b] April discovers that the quadratic polynomial $x^2 + 5x + 3$ has distinct roots $a$ and $b$. She also discovers that the quadratic polynomial $x^2 + 7x + 4$ has distinct roots $c$ and $d$. Compute $$ac + bc + bd + ad + a + b.$$ [b]p4.[/b] A rectangular picture frame that has a $2$ inch border can exactly fit a $10$ by $7$ inch photo. What is the total area of the frame's border around the photo, in square inches? [b]p5.[/b] Compute the median of the positive divisors of $9999$. [b]p6.[/b] Kaity only eats bread, pizza, and salad for her meals. However, she will refuse to have salad if she had pizza for the meal right before. Given that she eats $3$ meals a day (not necessarily distinct), in how many ways can we arrange her meals for the day? [b]p7.[/b] A triangle has side lengths $3$, $4$, and $x$, and another triangle has side lengths $3$, $4$, and $2x$. Assuming both triangles have positive area, compute the number of possible integer values for $x$. [b]p8.[/b] In the diagram below, the largest circle has radius $30$ and the other two white circles each have a radius of $15$. Compute the radius of the shaded circle. [img]https://cdn.artofproblemsolving.com/attachments/c/1/9eaf1064b2445edb15782278fc9c6efd1440b0.png[/img] [b]p9.[/b] What is the remainder when $2022$ is divided by $9$? [b]p10.[/b] For how many positive integers $x$ less than $2022$ is $x^3 - x^2 + x - 1$ prime? [b]p11.[/b] A sphere and cylinder have the same volume, and both have radius $10$. The height of the cylinder can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p12.[/b] Amanda, Brianna, Chad, and Derrick are playing a game where they pass around a red flag. Two players "interact" whenever one passes the flag to the other. How many different ways can the flag be passed among the players such that (1) each pair of players interacts exactly once, and (2) Amanda both starts and ends the game with the flag? [b]p13.[/b] Compute the value of $$\dfrac{12}{1 + \dfrac{12}{1+ \dfrac{12}{1+...}}}$$ [b]p14.[/b] Compute the sum of all positive integers $a$ such that $a^2 - 505$ is a perfect square. [b]p15.[/b] Alissa, Billy, Charles, Donovan, Eli, Faith, and Gerry each ask Sara a question. Sara must answer exactly $5$ of them, and must choose an order in which to answer the questions. Furthermore, Sara must answer Alissa and Billy's questions. In how many ways can Sara complete this task? [b]p16.[/b] The integers $-x$, $x^2 - 1$, and $x3$ form a non-decreasing arithmetic sequence (in that order). Compute the sum of all possible values of $x^3$. [b]p17.[/b] Moor and his $3$ other friends are trying to split burgers equally, but they will have $2$ left over. If they find another friend to split the burgers with, everyone can get an equal amount. What is the fewest number of burgers that Moor and his friends could have started with? [b]p18.[/b] Consider regular dodecagon $ABCDEFGHIJKL$ below. The ratio of the area of rectangle $AFGL$ to the area of the dodecagon can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/8/3/c38c10a9b2f445faae397d8a7bc4c8d3ed0290.png[/img] [b]p19.[/b] Compute the remainder when $3^{4^{5^6}}$ is divided by $4$. [b]p20.[/b] Fred is located at the middle of a $9$ by $11$ lattice (diagram below). At every second, he randomly moves to a neighboring point (left, right, up, or down), each with probability $1/4$. The probability that he is back at the middle after exactly $4$ seconds can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/c/f8e092e60f568ab7b28964d23b2ee02cdba7ad.png[/img] PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let a cube of side $1$ be given. Prove that there exists a point $A$ on the surface $S$ of the cube such that every point of $S$ can be joined to $A$ by a path on $S$ of length not exceeding $2$. Also prove that there is a point of $S$ that cannot be joined with $A$ by a path on $S$ of length less than $2$.

Let $ n \geq 3$ be a positive integer and let $ m \geq 2^{n\minus{}1}\plus{}1$. Prove that for each family of nonzero distinct subsets $ (A_j)_{j \in \overline{1, m}}$ of $ \{1, 2, ..., n\}$ there exist $ i$, $ j$, $ k$ such that $ A_i \cup A_j \equal{} A_k$.