In a circle there are $2019$ plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from $1$ to $16$, and Vasya moves the specified cake to the specified number of check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some $k$ pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest $k$ Petya can succeed?

The boba shop sells four different types of milk tea, and William likes to get tea each weekday. If William refuses to have the same type of tea on successive days, how many different combinations could he get, Monday through Friday?

Jason and Valerie agree to meet for game night, which runs from $4:00$ PM to $5:00$ PM. Jason and Valerie each choose a random time from $4:00$ PM to $5:00$ PM to show up. If Jason arrives first, he will wait $20$ minutes for Valerie before leaving. If Valerie arrives first, she will wait $10$ minutes for Jason before leaving. What is the probability that Jason and Valerie successfully meet each other for game night?

Poldavia is a strange kingdom. Its currency unit is the bourbaki and there exist only two types of coins: gold ones and silver ones. Each gold coin is worth $ n$ bourbakis and each silver coin is worth $ m$ bourbakis ($ n$ and $ m$ are positive integers). Using gold and silver coins, it is possible to obtain sums such as 10000 bourbakis, 1875 bourbakis, 3072 bourbakis, and so on. But Poldavia’s monetary system is not as strange as it seems: [b](a)[/b] Prove that it is possible to buy anything that costs an integral number of bourbakis, as long as one can receive change. [b](b)[/b] Prove that any payment above $ mn\minus{}2$ bourbakis can be made without the need to receive change.

A nonempty word is called pronounceable if it alternates in vowels (A, E, I, O, U) and consonants (all other letters) and it has at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off. [i]Proposed by Australia[/i]

On a $n \times n$ grid, each edge are written with $=$ or $\neq$. We need to filled every cells with color black or white. Find the largest constant $k$, such that for every $n>777771449$ and any layout of $=$ and $\neq$, we can always find a way to colored every cells, such that at least $k \cdot 2n(n-1)$ neighboring cells, there colors conform to the symbols on the edge. (Namely, two cells are filled with the same color if $=$ was written on their edge; two cells are filled with different colors if $\neq$ was written on their edge) [i]Proposed by chengbilly & sn6dh[/i]

Let $x_0,x_1,\dots, x_{n-1}$ be real numbers such that $0<|x_0|<|x_1|<\dots<|x_{n-1}|$. We will write the sum of the elements of each one of the $2^n$ subsets of $\{x_0,x_1,\dots,x_{n-1}\}$ in a paper. Prove that the $2^n$ written numbers are consecutive elements of a arithmetic progression if and only if the ratios $$|\frac{x_i}{x_j}|, 0\leq j<i\leq n-1$$ are equal(s) to the ratio(s) obtained with the numbers $2^0,2^1,\dots,2^{n-1}$. Note: The sum of the elements of the empty set is $0$.

Chameleons of five colors live on the island. When one chameleon bites another, the color of bitten chameleon changes to one of these five colors according to some rule, and the new color depends only on the color of the bitten and the color of the bitting. It is known that $2023$ red chameleons can agree on a sequence of bites between themselves, after which they will all turn blue. What is the smallest $k$ that can guarantee that $k$ red chameleons, biting only each other, can turn blue? (For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green one bites a red one, the bitten one remains red, that is, "changes color to red"; if red bites red, the bitten one changes color to yellow, etc. The rules for changing colors may be different.)

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.

How many paths are there from $A$ to $B$ that consist of $5$ horizontal segments and $5$ vertical segments of length $1$ each? (see figure) [img]https://cdn.artofproblemsolving.com/attachments/4/2/5b476ca2a232fc67fb2e2f6bb06111cab60692.png[/img]

Let us say that a deck of $52$ cards is arranged in a “regular” way if the ace of spades is on the very top of the deck and any two adjacent cards are either of the same value or of the same suit (top and bottom cards regarded adjacent as well). Prove that the number of ways to arrange a deck in regular way is a) divisible by $12!$ (3) b) divisible by $13!$ (5)

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

On an infinite white checkered sheet, a square $Q$ of size $12$ × $12$ is selected. Petya wants to paint some (not necessarily all!) cells of the square with seven colors of the rainbow (each cell is just one color) so that no two of the $288$ three-cell rectangles whose centers lie in $Q$ are the same color. Will he succeed in doing this? (Two three-celled rectangles are painted the same if one of them can be moved and possibly rotated so that each cell of it is overlaid on the cell of the second rectangle having the same color.) (Bogdanov. I)

There are $n^2$ empty boxes, each with a square base. The height and width of each box are integers between $1$ and $n$ inclusive, and no two boxes are identical. One box [i]fits inside[/i] another if its height and width are both smaller, and additionally, one of its dimensions is at least $2$ units smaller. In this way, we can form sequences of boxes (the first inside the second, the second inside the third, and so on). We place each of these sequences on a different shelf. How many shelves are needed to store all the boxes, with certainty?

For dessert, Melinda eats a spherical scoop of ice cream with diameter $2$ inches. She prefers to eat her ice cream in cube-like shapes, however. She has a special machine which, given a sphere placed in space, cuts it through the planes $x = n$, $y = n$, and $z = n$ for every integer $n$ (not necessarily positive). Melinda centers the scoop of ice cream uniformly at random inside the cube $0 \le x, y,z \le 1$, and then cuts it into pieces using her machine. What is the expected number of pieces she cuts the ice cream into?

Laura has $2010$ lamps connected with $2010$ buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible selection.) Richard always presses the buttons simultaneously, so the lamps are lit simultaneously, too. a) If Richard chooses the buttons to be pressed, what is the maximum number of different combinations of buttons he can press until Laura can assign the buttons to the lamps correctly? b) Supposing that Laura will choose the combinations of buttons to be pressed, what is the minimum number of attempts she has to do until she is able to associate the buttons with the lamps in a correct way?

Sean enters a classroom in the Memorial Hall and sees a $1$ followed by $2020$ $0$'s on the blackboard. As he is early for class, he decides to go through the digits from right to left and independently erase the $n$th digit from the left with probability $\frac{n-1}{n}$. (In particular, the $1$ is never erased.) Compute the expected value of the number formed from the remaining digits when viewed as a base-$3$ number. (For example, if the remaining number on the board is $1000$, then its value is $27$.)

Let $n_1, n_2, \cdots , n_{51}$ be distinct natural numbers each of which has exactly $2023$ positive integer factors. For instance, $2^{2022}$ has exactly $2023$ positive integer factors $1,2, 2^{2}, 2^{3}, \cdots 2^{2021}, 2^{2022}$. Assume that no prime larger than $11$ divides any of the $n_{i}$'s. Show that there must be some perfect cube among the $n_{i}$'s.

There are $n(n\ge 8)$ airports, some of which have one-way direct routes between them. For any two airports $a$ and $b$, there is at most one one-way direct route from $a$ to $b$ (there may be both one-way direct routes from $a$ to $b$ and from $b$ to $a$). For any set $A$ composed of airports $(1\le | A| \le n-1)$, there are at least $4\cdot \min \{|A|,n-|A| \}$ one-way direct routes from the airport in $A$ to the airport not in $A$. Prove that: For any airport $x$, we can start from $x$ and return to the airport by no more than $\sqrt{2n}$ one-way direct routes.

For all nonempty finite sets of point $S$ on the plane satisfying: $|S|$ is even and for all partitions of $S$ into two subsets $A,B$ of equal size, there is a reflection that maps $A$ to $B$.

[b]p1.[/b] Catherine's teacher thinks of a number and asks her to subtract $5$ and then multiply the result by $6$. Catherine accidentally switches the numbers by subtracting 6 and multiplying by $5$ to get $30$. If Catherine had not swapped the numbers, what would the correct answer be? [b]p2.[/b] At Acton Boxborough Regional High School, desks are arranged in a rectangular grid-like configuration. In order to maintain proper social distancing, desks are required to be at least 6 feet away from all other desks. Assuming that the size of the desks is negligible, what is the maximum number of desks that can fit in a $25$ feet by $25$ feet classroom? [b]p3.[/b] Joshua hates writing essays for homework, but his teacher Mr. Meesh assigns two essays every $3$ weeks. However, Mr. Meesh favors Joshua, so he allows Joshua to skip one essay out of every $4$ that are assigned. How many essays does Joshua have to write in a $24$-week school year? [b]p4.[/b] Libra likes to read, but she is easily distracted. If a page number is even, she reads the page twice. If a page number is an odd multiple of three, she skips it. Otherwise, she reads the page exactly once. If Libra's book is $405$ pages long, how many pages in total does she read if she starts on page $1$? (Reading the same page twice counts as two pages.) [b]p5.[/b] Let the GDP of an integer be its Greatest Divisor that is Prime. For example, the GDP of $14$ is $7$. Find the largest integer less than $100$ that has a GDP of $3$. [b]p6.[/b] As has been proven by countless scientific papers, the Earth is a flat circle. Bob stands at a point on the Earth such that if he walks in a straight line, the maximum possible distance he can travel before he falls off is $7$ miles, and the minimum possible distance he can travel before he falls off is $3$ miles. Then the Earth's area in square miles is $k\pi$ for some integer $k$. Compute $k$. [b]p7.[/b] Edward has $2$ magical eggs. Every minute, each magical egg that Edward has will double itself. But there's a catch. At the end of every minute, Edward's brother Eliot will come outside and smash one egg on his forehead, causing Edward to lose that egg permanently. For example, starting with $2$ eggs, after one minute there will be $3$ eggs, then $5$, $9$, and so on. After $1$ hour, the number of eggs can be expressed as $a^b + c$ for positive integers $a$, $b$, $c$ where $a > 1$, and $a$ and $c$ are as small as possible. Find $a + b + c$. [b]p8.[/b] Define a sequence of real numbers $a_1$, $a_2$, $a_3$, $..$, $a_{2019}$, $a_{2020}$ with the property that $a_n =\frac{a_{n-1} + a_n + a_{n+1}}{3}$ for all $n = 2$, $3$, $4$, $5$,$...$, $2018$, $2019$. Given that $a_1 = 1$ and $a_{1000} = 1999$, find $a_{2020}$. [b]p9.[/b] In $\vartriangle ABC$ with $AB = 10$ and $AC = 12$, points $D$ and $E$ lie on sides $\overline{AB}$ and $\overline{AC}$, respectively, such that $AD = 4$ and $AE = 5$. If the area of quadrilateral $BCED$ is $40$, find the area of $\vartriangle ADE$. [b]p10.[/b] A positive integer is called powerful if every prime in its prime factorization is raised to a power greater than or equal to $2$. How many positive integers less than 100 are powerful? [b]p11.[/b] Let integers $A,B < 10, 000$ be the populations of Acton and Boxborough, respectively. When $A$ is divided by $B$, the remainder is $1$. When $B$ is divided by $A$, the remainder is $2020$. If the sum of the digits of $A$ is $17$, find the total combined population of Acton and Boxborough. [b]p12.[/b] Let $a_1$, $a_2$, $...$, $a_n$ be an increasing arithmetic sequence of positive integers. Given $a_n - a_1 = 20$ and $a^2_n - a^2_{n-1} = 63$, find the sum of the terms in the arithmetic sequence. [b]p13.[/b] Bob rolls a cubical, an octahedral and a dodecahedral die ($6$, $8$ and $12$ sides respectively) numbered with the integers from $1$ to $6$, $1$ to $8$ and $1$ to $12$ respectively. If the probability that the sum of the numbers on the cubical and octahedral dice equals the number on the dodecahedral die can be written as $\frac{m}{n}$ , where $m, n$ are relatively prime positive integers, compute $n - m$. [b]p14.[/b] Let $\vartriangle ABC$ be inscribed in a circle with center $O$ with $AB = 13$, $BC = 14$, $AC = 15$. Let the foot of the perpendicular from $A$ to BC be $D$ and let $AO$ intersect $BC$ at $E$. Given the length of $DE$ can be expressed as $\frac{m}{n}$ where $m$, $n$ are relatively prime positive integers, find $m + n$. [b]p15.[/b] The set $S$ consists of the first $10$ positive integers. A collection of $10$ not necessarily distinct integers is chosen from $S$ at random. If a particular number is chosen more than once, all but one of its occurrences are removed. Call the set of remaining numbers $A$. Let $\frac{a}{b}$ be the expected value of the number of the elements in $A$, where $a, b$ are relatively prime positive integers. Find the reminder when $a + b$ is divided by $1000$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be a string of $99$ characters, $66$ of which are $A$ and $33$ are $B$. We call $S$ [i]good[/i] if, for each $n$ such that $1\le n \le 99$, the sub-string made from the first $n$ characters of $S$ has an odd number of distinct permutations. How many good strings are there? Which strings are good?

Alice and Bob play a game. First, Alice secretly picks a finite set $S$ of lattice points in the Cartesian plane. Then, for every line $\ell$ in the plane which is horizontal, vertical, or has slope $+1$ or $-1$, she tells Bob the number of points of $S$ that lie on $\ell$. Bob wins if he can determine the set $S$. Prove that if Alice picks $S$ to be of the form \[S = \{(x, y) \in \mathbb{Z}^2 \mid m \le x^2 + y^2 \le n\}\] for some positive integers $m$ and $n$, then Bob can win. (Bob does not know in advance that $S$ is of this form.) [i]Proposed by Mark Sellke[/i]

Find all family $\mathcal{F}$ of subsets of $[n]$ such that for any nonempty subset $X\subseteq [n]$, exactly half of the elements $A\in \mathcal{F}$ satisfies that $|A\cap X|$ is even.