Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.) [i](Poland) Marek Cygan[/i]

Bengt wants to put out crosses and rings in the squares of an $n \times n$-square, so that it is exactly one ring and exactly one cross in each row and in each column, and no more than one symbol in each box. Mona wants to stop him by setting a number in advance ban on crosses and a number of bans on rings, maximum one ban in each square. She want to use as few bans as possible of each variety. To succeed in preventing Bengt, how many prohibitions she needs to use the least of the kind of prohibitions she uses the most of?

There are three stacks of tokens on the table: the first contains $a,$ the second contains $b,$ and the third contains $c$ where $a \ge b \ge c.$ Players $A$ and $B$ take turns playing a game of swapping tokens. $A$ starts first. On each turn, the player who gets his turn chooses two stacks, then takes at least one token from the stack with the lowest number of tokens and places them on the stack with the highest number of tokens. If the number of tokens in the two piles he/she chooses is equal, then he/she takes at least one token from any of them and puts it in the other. If only one pile remains after a player's move, that player is considered a winner. At what values of $a, b, c$ who has the winning strategy ($A$ or $B$)?

Alberto and Barbara play the following game. Initially, there are some piles of coins on a table. Each player in turn, starting with Albert, performs one of the two following ways: 1) take a coin from an arbitrary pile; 2) select a pile and divide it into two non-empty piles. The winner is the player who removes the last coin on the table. Determine which player has a winning strategy with respect to the initial state.

Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have. (Two cells of the grid are adjacent if they have a common vertex)

In a square of sidelength $7$, $51$ points are given. Show that there's a disk of radius $1$ covering at least $3$ of these points.

How many sequences $a_1$, $a_2$, $...$,$a_8$ of zeroes and ones have $a_1a_2 + a_2a_3 +...+ a_7a_8 = 5$?

Let $n>1$ be a given integer. In a deck of cards the cards are of $n$ different suites and $n$ different values, and for each pair of a suite and a value there is exactly one such card. We shuffle the deck and distribute the cards among $n$ players giving each player $n$ cards. The players' goal is to choose a way to sit down around a round table so that they will be able to do the following: the first player puts down an arbitrary card, and then each consecutive player puts down a card that has a different suite and different value compared to the previous card that was put down on the table. For which $n$ is it possible that the cards were distributed in such a way that the players cannot achieve their goal? (The players work together, and they can see each other's cards.) Proposed by [i]Anett Kocsis[/i], Budapest

How many permutations of the set $\{B, M, T, 2,0\}$ do not have $B$ as their first element?

A rectangular parallelepiped painted blue is cut into $1 \times 1\times 1$ cubes. Find the possible dimensions if the number of cubes without blue faces is equal to one-third of the total number of cubes. [b]Note:[/b] A [i]rectangular parallelepiped[/i] is a solid with $6$ faces, all of which are rectangles (or squares).

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [b]C1.[/b] Mr. Pham flips $2018$ coins. What is the difference between the maximum and minimum number of heads that can appear? [b]C2 / G1.[/b] Brandon wants to maximize $\frac{\Box}{\Box} +\Box$ by placing the numbers $1$, $2$, and $3$ in the boxes. If each number may only be used once, what is the maximum value attainable? [b]C3.[/b] Guang has $10$ cents consisting of pennies, nickels, and dimes. What are all the possible numbers of pennies he could have? [b]C4.[/b] The ninth edition of Campbell Biology has $1464$ pages. If Chris reads from the beginning of page $426$ to the end of page$449$, what fraction of the book has he read? [b]C5 / G2.[/b] The planet Vriky is a sphere with radius $50$ meters. Kyerk starts at the North Pole, walks straight along the surface of the sphere towards the equator, runs one full circle around the equator, and returns to the North Pole. How many meters did Kyerk travel in total throughout his journey? [b]C6 / G3.[/b] Mr. Pham is lazy and decides Stan’s quarter grade by randomly choosing an integer from $0$ to $100$ inclusive. However, according to school policy, if the quarter grade is less than or equal to $50$, then it is bumped up to $50$. What is the probability that Stan’s final quarter grade is $50$? [b]C7 / G5.[/b] What is the maximum (finite) number of points of intersection between the boundaries of a equilateral triangle of side length $1$ and a square of side length $20$? [b]C8.[/b] You enter the MBMT lottery, where contestants select three different integers from $1$ to $5$ (inclusive). The lottery randomly selects two winning numbers, and tickets that contain both of the winning numbers win. What is the probability that your ticket will win? [b]C9 / G7.[/b] Find a possible solution $(B, E, T)$ to the equation $THE + MBMT = 2018$, where $T, H, E, M, B$ represent distinct digits from $0$ to $9$. [b]C10.[/b] $ABCD$ is a unit square. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $AD$. $DE$ and $CF$ meet at $G$. Find the area of $\vartriangle EFG$. [b]C11.[/b] The eight numbers $2015$, $2016$, $2017$, $2018$, $2019$, $2020$, $2021$, and $2022$ are split into four groups of two such that the two numbers in each pair differ by a power of $2$. In how many different ways can this be done? [b]C12 / G4.[/b] We define a function f such that for all integers $n, k, x$, we have that $$f(n, kx) = k^n f(n, x) and f(n + 1, x) = xf(n, x).$$ If $f(1, k) = 2k$ for all integers $k$, then what is $f(3, 7)$? [b]C13 / G8.[/b] A sequence of positive integers is constructed such that each term is greater than the previous term, no term is a multiple of another term, and no digit is repeated in the entire sequence. An example of such a sequence would be $4$, $79$, $1035$. How long is the longest possible sequence that satisfies these rules? [b]C14 / G11.[/b] $ABC$ is an equilateral triangle of side length $8$. $P$ is a point on side AB. If $AC +CP = 5 \cdot AP$, find $AP$. [b]C15.[/b] What is the value of $(1) + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + ... + 49 + 50)$? [b]G6.[/b] An ant is on a coordinate plane. It starts at $(0, 0)$ and takes one step each second in the North, South, East, or West direction. After $5$ steps, what is the probability that the ant is at the point $(2, 1)$? [b]G10.[/b] Find the set of real numbers $S$ so that $$\prod_{c\in S}(x^2 + cxy + y^2) = (x^2 - y^2)(x^{12} - y^{12}).$$ [b]G12.[/b] Given a function $f(x)$ such that $f(a + b) = f(a) + f(b) + 2ab$ and $f(3) = 0$, find $f\left( \frac12 \right)$. [b]G13.[/b] Badville is a city on the infinite Cartesian plane. It has $24$ roads emanating from the origin, with an angle of $15$ degrees between each road. It also has beltways, which are circles centered at the origin with any integer radius. There are no other roads in Badville. Steven wants to get from $(10, 0)$ to $(3, 3)$. What is the minimum distance he can take, only going on roads? [b]G14.[/b] Team $A$ and Team $B$ are playing basketball. Team A starts with the ball, and the ball alternates between the two teams. When a team has the ball, they have a $50\%$ chance of scoring $1$ point. Regardless of whether or not they score, the ball is given to the other team after they attempt to score. What is the probability that Team $A$ will score $5$ points before Team $B$ scores any? [b]G15.[/b] The twelve-digit integer $$\overline{A58B3602C91D},$$ where $A, B, C, D$ are digits with $A > 0$, is divisible by $10101$. Find $\overline{ABCD}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $32$ students in the class with $10$ interesting group. Each group contains exactly $16$ students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their $interests-disparity$. Define $S$ as the sum of the $interests-disparity$ of all the couples, $\binom{32}{2}\left ( =\: 496 \right )$ ones in total. Determine the minimal possible value of $S$.

16 NHL teams in the first playoff round divided in pairs and to play series until 4 wins (thus the series could finish with score 4-0, 4-1, 4-2, or 4-3). After that 8 winners of the series play the second playoff round divided into 4 pairs to play series until 4 wins, and so on. After all the final round is over, it happens that $k$ teams have non-negative balance of wins (for example, the team that won in the first round with a score of 4-2 and lost in the second with a score of 4-3 fits the condition: it has $4+3=7$ wins and $2+4=6$ losses). Find the least possible $k$.

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

Suppose that n numbers $x_1, x_2, . . . , x_n$ are chosen randomly from the set $\{1, 2, 3, 4, 5\}$. Prove that the probability that $x_1^2+ x_2^2 +\cdots+ x_n^2 \equiv 0 \pmod 5$ is at least $\frac 15.$

The front row of a movie theatre contains $45$ seats. [list] [*] (a) If $42$ people are sitting in the front row, prove that there are $10$ consecutive seats that are all occupied. [*] (b) Show that this conclusion doesn’t necessarily hold if only $41$ people are sitting in the front row.[/list]

A certain town is represented as an infinite plane, which is divided by straight lines into squares. The lines are streets, while the squares are blocks. Along a certain street there stands a policeman on each $100$th intersection . Somewhere in the town there is a bandit , whose position and speed are unknown, but he can move only along the streets. The aim of the police is to see the bandit . Does there exist an algorithm available to the police to enable them to achieve their aim? (A. Andjans, Riga)

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

In a ballroom, seven gentlemen $A, B, C, D, E, F$ and $G$ sit directly across from seven queens $a, b, c, d, e, f$ and $g$ in random order. When the gentlemen rise and walks across the dance floor to bow to each of their ladies, someone notices that at least two of the men travel equally long distances. Will it always be like this? The figure shows an example. In the example, $|Bb| =|Ee|$ and $|Dd|=|Cc|$. [img]https://cdn.artofproblemsolving.com/attachments/8/3/1e18a30b1e9acc90b24210fc7991b58062a69f.png[/img]

Ten children have several bags of candies. The children begin to divide these candies among them. They take turns picking their shares of candies from each bag, and leave just after that. The size of the share is determined as follows: the current number of candies in the bag is divided by the number of remaining children (including the one taking the turn). If the remainder is nonzero than the quotient is rounded to the lesser integer. Is it possible that all the children receive different numbers of candies if the total number of bags is: a) 8 ; 6) 99 ? Alexey Glebov

Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist $5$ different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box

Let $\mathcal{G}$ be a simple graph with $n$ vertices and $m$ edges such that no two cycles share an edge. Prove that $2m < 3n$. [i]Note[/i]: A [i]simple graph[/i] is a graph with at most one edge between any two vertices and no edges from any vertex to itself. A [i]cycle[/i] is a sequence of distinct vertices $v_1, \dots, v_n$ such that there is an edge between any two consecutive vertices, and between $v_n$ and $v_1$. [i]Proposed by Ethan Tan[/i]

Let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space. Determine the smallest number $g(n)$ of a points of a set in $\mathbb{R}^n$ such that every point in $\mathbb{R}^n$ is an irrational distance from at least one point in that set.

Let $m, n$ be positive integers $(m, n>=2)$. Given an $n$-element set $A$ of integers $(A=\{a_1,a_2,\cdots ,a_n\})$, for each pair of elements $a_i, a_j(j>i)$, we make a difference by $a_j-a_i$. All these $C^2_n$ differences form an ascending sequence called “derived sequence” of set $A$. Let $\bar{A}$ denote the derived sequence of set $A$. Let $\bar{A}(m)$ denote the number of terms divisible by $m$ in $\bar{A}$ . Prove that $\bar{A}(m)\ge \bar{B}(m)$ where $A=\{a_1,a_2,\cdots ,a_n\}$ and $B=\{1,2,\cdots ,n\}$.

Two-way flights are operated between $80$ cities in such a way that each city is connected to at least $7$ other cities by a direct flight and any two cities are connected by a finite sequence of flights. Find the smallest $k$ such that for any such arrangement of flights it is possible to travel from any city to any other city by a sequence of at most $k$ flights.