Billiam is distributing his ample supply of balls among an ample supply of boxes. He distributes the balls as follows: he places a ball in the first empty box, and then for the greatest positive integer n such that all $n$ boxes from box $1$ to box $n$ have at least one ball, he takes all of the balls in those $n$ boxes and puts them into box $n +1$. He then repeats this process indefinitely. Find the number of repetitions of this process it takes for one box to have at least $2022$ balls.

Some books are placed on each other. Someone first, reverses the upper book. Then he reverses the $2$ upper books. Then he reverses the $3$ upper books and continues like this. After he reversed all the books, he starts this operation from the first. Prove that after finite number of movements, the books become exactly like their initial configuration.

[b]p1.[/b] Chad lives on the third floor of an apartment building with ten floors. He leaves his room and goes up two floors, goes down four floors, goes back up five floors, and finally goes down one floor, where he finds Jordan's room. On which floor does Jordan live? [b]p2.[/b] A real number $x$ satisfies the equation $2014x + 1337 = 1337x + 2014$. What is $x$? [b]p3.[/b] Given two points on the plane, how many distinct regular hexagons include both of these points as vertices? [b]p4.[/b] Jordan has six different files on her computer and needs to email them to Chad. The sizes of these files are $768$, $1024$, $2304$, $2560$, $4096$, and $7680$ kilobytes. Unfortunately, the email server holds a limit of $S$ kilobytes on the total size of the attachments per email, where $S$ is a positive integer. It is additionally given that all of the files are indivisible. What is the maximum value of S for which it will take Jordan at least three emails to transmit all six files to Chad? [b]p5.[/b] If real numbers $x$ and $y$ satisfy $(x + 2y)^2 + 4(x + 2y + 2 - xy) = 0$, what is $x + 2y$? [b]p6.[/b] While playing table tennis against Jordan, Chad came up with a new way of scoring. After the first point, the score is regarded as a ratio. Whenever possible, the ratio is reduced to its simplest form. For example, if Chad scores the first two points of the game, the score is reduced from $2:0$ to $1:0$. If later in the game Chad has $5$ points and Jordan has $9$, and Chad scores a point, the score is automatically reduced from $6:9$ to $2:3$. Chad's next point would tie the game at $1:1$. Like normal table tennis, a player wins if he or she is the first to obtain $21$ points. However, he or she does not win if after his or her receipt of the $21^{st}$ point, the score is immediately reduced. Chad and Jordan start at $0:0$ and finish the game using this rule, after which Jordan notes a curiosity: the score was never reduced. How many possible games could they have played? Two games are considered the same if and only if they include the exact same sequence of scoring. [b]p7.[/b] For a positive integer $m$, we define $m$ as a factorial number if and only if there exists a positive integer $k$ for which $m = k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$. We define a positive integer $n$ as a Thai number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers. What is the sum of the five smallest Thai numbers? [b]p8.[/b] Chad and Jordan are in the Exeter Space Station, which is a triangular prism with equilateral bases. Its height has length one decameter and its base has side lengths of three decameters. To protect their station against micrometeorites, they install a force field that contains all points that are within one decameter of any point of the surface of the station. What is the volume of the set of points within the force field and outside the station, in cubic decameters? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integer numbers $n\ge 4$ which satisfy the following condition: from every $n$ different $3$-element subsets of $n$-element set it is possible to choose $2$ subsets, which have exactly one element in common.

There are 100 persons in a hall, everyone knowing at least 66 of the others. Prove that there is a case in which among any four some two don’t know each other.

A dog has infinitely many pieces of meat, but on each piece of meat there is a fly. At each move, the dog does the following: [list=1] [*] He eats a piece of meat together with all flies lying on it; [*] He moves a fly from a piece of meat to another. [/list] The dog doesn't want to eat more than one milion flies. Prove that he cannot ensure that each piece of meat is eaten at some point. [i]Proposed by I. Mitrofanov[/i]

[b]p1.[/b] You are given three buckets with a capacity to hold $8$, $5$, and $3$ quarts of water, respectively. Initially, the first bucket is filled with $8$ quarts of water, while the remaining two buckets are empty. There are no markings on the buckets, so you are only allowed to empty a bucket into another one or to fill a bucket to its capacity using the water from one of the other buckets. (a) Describe a procedure by which we can obtain exactly $6$ quarts of water in the first bucket. (b) Describe a procedure by which we can obtain exactly $4$ quarts of water in the first bucket. [b]p2.[/b] A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. In each case below, give explicitly the coordinates of the vertices of a lattice triangle $T$ that satisfies the stated properties. (a) The area of $T$ is $1/2$ and two sides of $T$ have length greater than $2011$. (b) The area of $T$ is $1/2$ and the three sides of $T$ each have length greater than $2011$. [b]p3.[/b] Alice and Bob play several rounds of a game. In the $n$-th round, where $n = 1, 2, 3, ...$, the loser pays the winner $2^{n-1}$ dollars (there are no ties). After $40$ rounds, Alice has a profit of $\$2011$ (and Bob has lost $\$2011$). How many rounds of the game did Alice win, and which rounds were they? Justify your answer. [b]p4.[/b] Each student in a school is assigned a $15$-digit ID number consisting of a string of $3$’s and $7$’s. Whenever $x$ and $y$ are two distinct ID numbers, then $x$ and $y$ differ in at least three entries. Show that the number of students in the school is less than or equal to $2048$. [b]p5.[/b] A triangle $ABC$ has the following property: there is a point $P$ in the plane of $ABC$ such that the triangles $PAB$, $PBC$ and $PCA$ all have the same perimeter and the same area. Prove that: (a) If $P$ is not inside the triangle $ABC$, then $ABC$ is a right-angled triangle. (b) If $P$ is inside the triangle $ABC$, then $ABC$ is an equilateral triangle. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Zaineb makes a large necklace from beads labeled $290, 291, \ldots, 2023$. She uses each bead exactly once, arranging the beads in the necklace any order she likes. Prove that no matter how the beads are arranged, there must be three beads in a row whose labels are the side lengths of a triangle.

Prove that among any $n$ positive integers one can always find some (at least one) whose sum is divisible by $n$.

A student wants to make his birthday party special this year. He wants to organize it such that among any groups of $4$ persons at the party there is one that is friends with exactly another person in the group. Find the largest number of his friends that he can possibly invite at the party.

In a community of more than six people each member exchanges letters with exactly three other members of the community. Show that the community can be partitioned into two nonempty groups so that each member exchanges letters with at least two members of the group he belongs to.

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 1$)?

$N$ unfair coins (with heads and tails on the sides) are thrown, with the $k^{th}$ coin has got a chance of $\frac{1}{2k + 1}$ to land on tails.How high is the probability that an odd number of coins will show tails?

Consider the set $S$ of lattice points with positive coordinates in the plane. For each point $P(a,b)$ from $S$, we draw a segment between it and each of the points in the set \[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\] Show that there is no colouring of the points in $S$ with a finite number of colours such that every two points joined by a segment are coloured with different colours. [i]Ioan Tomescu[/i]

During a day $2016$ customers visited the store. Every customer has been only once at the store(a customer enters the store,spends some time, and leaves the store). Find the greatest integer $k$ that makes the following statement always true. We can find $k$ customers such that either all of them have been at the store at the same time, or any two of them have not been at the same store at the same time.

We have 11 boxes. On a move, we can choose 10 of them and put one ball in each of the boxes chosen. Two players move alternately. The one who gets a box of 21 balls wins. Which of the two players has winning strategy?

In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points. A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on. Find the minimum value of a [i]good[/i] number $m$.

[b]p1[/b]. Points $P_1,P_2,P_3,... ,P_n$ lie on a plane such that $P_aP_b = 1$,$P_cP_d = 2$, and $P_eP_f = 2018$ for not necessarily distinct indices $a,b,c,d,e, f \in \{1, 2,... ,n\}$. Find the minimum possible value of $n$. [b]p2.[/b] Find the coefficient of the $x^2y^4$ term in the expansion of $(3x +2y)^6$. [b]p3.[/b] Find the number of positive integers $n < 1000$ such that $n$ is a multiple of $27$ and the digit sum of $n$ is a multiple of $11$. [b]p4.[/b] How many times do the minute hand and hour hand of a $ 12$-hour analog clock overlap in a $366$-day leap year? [b]p5.[/b] Find the number of ordered triples of integers $(a,b,c)$ such that $(a +b)(b +c)(c + a) = 2018$. [b]p6.[/b] Let $S$ denote the set of the first $2018$ positive integers. Call the score of a subset the sum of its maximal element and its minimal element. Find the sum of score $(x)$ over all subsets $s \in S$ [b]p7.[/b] How many ordered pairs of integers $(a,b)$ exist such that $1 \le a,b \le 20$ and $a^a$ divides $b^b$? [b]p8.[/b] Let $f$ be a function such that for every non-negative integer $p$, $f (p)$ equals the number of ordered pairs of positive integers $(a,n)$ such that $a^n = a^p \cdot n$. Find $\sum^{2018}_{p=0}f (p)$. [b]p9.[/b] A point $P$ is randomly chosen inside a regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$. What is the probability that the projections of $P$ onto the lines $\overleftrightarrow{A_i A_{i+1}}$ for $i = 1,2,... ,8$ lie on the segments $\overline{A_iA_{i+1}}$ for $i = 1,2,... ,8$ (where indices are taken $mod \,\, 8$)? [b]p10. [/b]A person keeps flipping an unfair coin until it flips $3$ tails in a row. The probability of it landing on heads is $\frac23$ and the probability it lands on tails is $\frac13$ . What is the expected value of the number of the times the coin flips? PS. You had better use hide for answers.

In a tennis tournament, any two of the $n$ participants played a match (the winner of a match gets $1$ point, the loser gets $0$). The scores after the tournament were $r_1\le r_2\le\ldots\le r_n$. A match between two players is called wrong if after it the winner has a score less than or equal to that of the loser. Consider the set $I=\{i|r_1\ge i\}$. Prove that the number of wrong matches is not less than $\sum_{i\in I}(r_i-i+1)$, and show that this value is realizable

In an $11 \times 11$ grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants to walk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in opposite directions). Compute the maximum number of doors he can go through in such a path.

Find all pairs of positive integers $(m, n)$, such that in a $m \times n$ table (with $m+1$ horizontal lines and $n+1$ vertical lines), a diagonal can be drawn in some unit squares (some unit squares may have no diagonals drawn, but two diagonals cannot be both drawn in a unit square), so that the obtained graph has an Eulerian cycle.

Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family.

Tom painted round fence which consists of $2n \ge6$ sections in such way that every section is painted in one of four colours. Then he repeats the following while it is possible: he chooses three neighbouring sections of distinct colours and repaints them into the fourth colour. For which $n$ Tom can repaint the fence in such way infinitely many times?

The permutation $\sigma$ consisting of four words $A,B,C,D$ has $f_{AB}(\sigma)$, the sum of the number of $B$ placed rightside of every $A$. We can define $f_{BC}(\sigma)$,$f_{CD}(\sigma)$,$f_{DA}(\sigma)$ as the same way too. For example, $\sigma=ACBDBACDCBAD$, $f_{AB}(\sigma)=3+1+0=4$, $f_{BC}(\sigma)=4$,$f_{CD}(\sigma)=6$, $f_{DA}(\sigma)=3$ Find the maximal value of $f_{AB}(\sigma)+f_{BC}(\sigma)+f_{CD}(\sigma)+f_{DA}(\sigma)$, when $\sigma$ consists of $2020$ letters for each $A,B,C,D$