Consider the set $S = \{1, 2, . . . , 2015\}$. How many ways are there to choose $2015$ distinct (possibly empty and possibly full) subsets $X_1, X_2, . . . , X_{2015}$ of $S$ such that $X_i$ is strictly contained in $X_{i+1}$ for all $1 \le i \le 2014$?

For a randomly selected permutation $ \mathbf{f} = (f_1,..., f_n) $ of the set $ \{1,\ldots, n\} $ let us denote by $ X(\mathbf{f}) $ the largest number $ k \leq n $ such that $ f_i < f_{ i+1} $ for all numbers $ i < k $. Prove that the expected value of the random variable $ X $ is $ \sum_{k=1}^n \frac{1}{k!} $.

Is it possible to place the numbers $0,1,2,\dots,9$ on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?

For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that $C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.

10000 children came to a camp; every of them is friend of exactly eleven other children in the camp (friendship is mutual). Every child wears T-shirt of one of seven rainbow's colours; every two friends' colours are different. Leaders demanded that some children (at least one) wear T-shirts of other colours (from those seven colours). Survey pointed that 100 children didn't want to change their colours [translator's comment: it means that any of these 100 children (and only them) can't change his (her) colour such that still every two friends' colours will be different]. Prove that some of other children can change colours of their T-shirts such that as before every two friends' colours will be different.

Let $A$ be the set of permutations $a=(a_1,a_2,…,a_n)$ of $M=\{1,2,…n\}$ with the following property: There doesn’t exist a subset $S$ of $M$ such that $a(S)=S$. For $\forall$ such permutation $a$ let $d(a)=\sum_{k=1}^n (a_k-k)^2$ . Determine the smallest value of $d(a)$.

Vulcan and Neptune play a turn-based game on an infinite grid of unit squares. Before the game starts, Neptune chooses a finite number of cells to be [i]flooded[/i]. Vulcan is building a [i]levee[/i], which is a subset of unit edges of the grid (called [i]walls[/i]) forming a connected, non-self-intersecting path or loop*. The game then begins with Vulcan moving first. On each of Vulcan’s turns, he may add up to three new walls to the levee (maintaining the conditions for the levee). On each of Neptune’s turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well. Prove that Vulcan can always, in a finite number of turns, build the levee into a closed loop such that all flooded cells are contained in the interior of the loop, regardless of which cells Neptune initially floods. ----- [size=75]*More formally, there must exist lattice points $\mbox{\footnotesize \(A_0, A_1, \dotsc, A_k\)}$, pairwise distinct except possibly $\mbox{\footnotesize \(A_0 = A_k\)}$, such that the set of walls is exactly $\mbox{\footnotesize \(\{A_0A_1, A_1A_2, \dotsc , A_{k-1}A_k\}\)}$. Once a wall is built it cannot be destroyed; in particular, if the levee is a closed loop (i.e. $\mbox{\footnotesize \(A_0 = A_k\)}$) then Vulcan cannot add more walls. Since each wall has length $\mbox{\footnotesize \(1\)}$, the length of the levee is $\mbox{\footnotesize \(k\)}$.[/size]

Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?

In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam. [hide=original wording]En un examen fueron propuestos 6 problemas. Cada problema fue resuelto por exactamente 1000 estudiantes, pero en ningun caso ha ocurrido que dos estudiantes en conjunto, hayan resuelto los 6 problemas. Determinar el menor numero de participantes que pudo haber en dicho exame[/hide]

In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most $100$ cities at distance exactly three from it. Prove that there is no city such that more than $2550$ other cities have distance exactly four from it.

A quadruple of integers $(a, b, c, d)$ is said good if $ad-bc=2020$. Two good quadruplets are said to be dissimilar if it is not possible to obtain one from the other using a finite number of applications of the following operations: $$(a,b,c,d) \rightarrow (-c,-d,a,b)$$ $$(a,b,c,d) \rightarrow (a,b,c+a,d+b)$$ $$(a,b,c,d) \rightarrow (a,b,c-a,d-b)$$ Let $A$ be a set of $k$ good quadruples, two by two dissimilar. Show that $k \leq 4284$.

All the natural numbers from $1$ to $1982$ are gathered in an array in an arbitrary order in computer's memory. The program looks through all the sequent pairs (first and second, second and third,...) and exchanges numbers in the pair, if the number on the lower place is greater than another. Then the program repeats the process, but moves from another end of the array. The number, that stand initially on the $100$-th place reserved its place. Find that number.

On a square board divided into $15 \times 15$ little squares there are $15$ rooks that do not attack each other. Then each rook makes one move like that of a knight. Prove that after this is done a pair of rooks will necessarily attack each other.

A teacher gives a multiple choice test to $15$ students and that each student answered each question. Each question had $5$ choices, but remarkably, no pair of students had more than $2$ answers in common. What is the maximum number of questions that could have been on the quiz?

Consider \(n^2\) unit squares in the \(xy\) plane centered at point \((i,j)\) with integer coordinates, \(1 \leq i \leq n\), \(1 \leq j \leq n\). It is required to colour each unit square in such a way that whenever \(1 \leq i < j \leq n\) and \(1 \leq k < l \leq n\), the three squares with centres at \((i,k),(j,k),(j,l)\) have distinct colours. What is the least possible number of colours needed?

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

On the plane 100 lines are drawn, among which there are no parallel lines. From any five of these lines, some three pass through one point. Prove that there are two points such that each line contains at least of of them.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]D16.[/b] The cooking club at Blair creates $14$ croissants and $21$ danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least two danishes. How many pastries must he choose to guarantee that he has one croissant and two danishes? [b]D17.[/b] Each digit in a $3$ digit integer is either $1, 2$, or $4$ with equal probability. What is the probability that the hundreds digit is greater than the sum of the tens digit and the ones digit? [b]D18 / Z11.[/b] How many two digit numbers are there such that the product of their digits is prime? [b]D19 / Z9.[/b] In the coordinate plane, a point is selected in the rectangle defined by $-6 \le x \le 4$ and $-2 \le y \le 8$. What is the largest possible distance between the point and the origin, $(0, 0)$? [b]D20 / Z10.[/b] The sum of two numbers is $6$ and the sum of their squares is $32$. Find the product of the two numbers. [u]Set 5[/u] [b]D21 / Z12.[/b] Triangle $ABC$ has area $4$ and $\overline{AB} = 4$. What is the maximum possible value of $\angle ACB$? [b]D22 / Z13.[/b] Let $ABCD$ be an iscoceles trapezoid with $AB = CD$ and M be the midpoint of $AD$. If $\vartriangle ABM$ and $\vartriangle MCD$ are equilateral, and $BC = 4$, find the area of trapezoid $ABCD$. [b]D23 / Z14.[/b] Let $x$ and $y$ be positive real numbers that satisfy $(x^2 + y^2)^2 = y^2$. Find the maximum possible value of $x$. [b]D24 / Z17.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$. [b]D25.[/b] The number $12ab9876543$ is divisible by $101$, where $a, b$ represent digits between $0$ and $9$. What is $10a + b$? [u]Set 6[/u] [b]D26 / Z26.[/b] For every person who wrote a problem that appeared on the final MBMT tests, take the number of problems they wrote, and then take that number’s factorial, and finally multiply all these together to get $n$. Estimate the greatest integer $a$ such that $2^a$ evenly divides $n$. [b]D27 / Z27.[/b] Circles of radius $5$ are centered at each corner of a square with side length $6$. If a random point $P$ is chosen randomly inside the square, what is the probability that $P$ lies within all four circles? [b]D28 / Z28.[/b] Mr. Rose’s evil cousin, Mr. Caulem, has teaches a class of three hundred bees. Every week, he tries to disrupt Mr. Rose’s $4$th period by sending three of his bee students to fly around and make human students panic. Unfortunately, no pair of bees can fly together twice, as then Mr. Rose will become suspicious and trace them back to Mr. Caulem. What’s the largest number of weeks Mr. Caulem can disrupt Mr. Rose’s class? [b]D29 / Z29. [/b]Two blind brothers Beard and Bored are driving their tractors in the middle of a field facing north, and both are $10$ meters west from a roast turkey. Beard, can turn exactly $0.7^o$ and Bored can turn exactly $0.2^o$ degrees. Driving at a consistent $2$ meters per second, they drive straight until they notice the smell of the turkey getting farther away, and then turn right and repeat until they get to the turkey. Suppose Beard gets to the Turkey in about $818.5$ seconds. Estimate the amount of time it will take Bored. [b]D30 / Z30.[/b] Let a be the probability that $4$ randomly chosen positive integers have no common divisor except for $1$. Estimate $300a$. Note that the integers $1, 2, 3, 4$ have no common divisor except for $1$. Remark. This problem is asking you to find $300 \lim_{n\to \infty} a_n$, if $a_n$ is defined to be the probability that $4$ randomly chosen integers from $\{1, 2, ..., n\}$ have greatest common divisor $1$. PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose an ordered set of $ ({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}}) $ real numbers, $n \ge 3 $. It is possible to replace the number $ {{a} _ {i}} $, $ i = \overline {2, \ n-1} $ by the number $ a_ {i} ^ {*} $ that $ {{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}} $. Let $ ({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}}) $ be the set with the largest sum of numbers that can be obtained from this, and $ ({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}}) $ is a similar set with the least amount. For the odd $n \ge 3 $ and set $ (1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1) $ find the values of the expressions $ {{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}} $ and $ {{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}} $.

In a conference, mathematicians from $11$ different countries participate and they have integer-valued ages between $27$ and $33$ years (including $27$ and $33$). There is at least one mathematician from each country, and there is at least one mathematician of each possible age between $27$ and $33$. Show that we can find at least five mathematicians $m_1, \ldots, m_5$ such that for any $i \in \{1, \ldots, 5 \}$ there are more mathematicians in the conference having the same age as $m_i$ than those having the same nationality as $m_i$. [i]Proposed by S. Muralidharan[/i]

At a strange party, each person knew exactly $22$ others. For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew. For any pair of people $X$ and $Y$ who did not know one another, there were exactly 6 other people that they both knew. How many people were at the party?

On a soccer tournament with $n\ge 3$ teams taking part, several matches are played in such a way that among any three teams, some two play a match. $(a)$ If $n=7$, find the smallest number of matches that must be played. $(b)$ Find the smallest number of matches in terms of $n$.

A nonempty set $A$ is called an [i]$n$-level-good [/i]set if $ A \subseteq \{1,2,3,\ldots,n\}$ and $|A| \le \min_{x\in A} x$ (where $|A|$ denotes the number of elements in $A$ and $\min_{x\in A} x$ denotes the minimum of the elements in $A$). Let $a_n$ be the number of $n$-level-good sets. Prove that for all positive integers $n$ we have $a_{n+2}=a_{n+1}+a_{n}+1$.

There is $500$ dollars in a bank. Two bank operations are allowed: to withdraw $300$ dollars from the bank or to deposit $198$ dollars into the bank. These operations can be repeated as many times as necessary but only the money that was initially in the bank can be used. What is the largest amount of money that can be borrowed from the bank? How can this be done? (AK Tolpygo)

Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: \[ \begin{array}{rccccccccc} \text{Row } 1: &&&&& a_{0,1} &&&& \smallskip\\ \text{Row } 2: &&&& a_{0,2} && a_{1,2} &&& \smallskip\\ &&& \vdots && \vdots && \vdots && \smallskip\\ \text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\ \text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n} \end{array} \] Call such a triangular array [i]stable[/i] if for every $0 \le i < j < k \le n$ we have \[ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. \] For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.