In a class of $14$ boys, each boy was asked how many classmates had the same first name. and how many colleagues had the same last name as them. The numbers $0, 1, 2, 3, 4, 5$ and $6$. Proves that there are two colleagues with the same first name and the same last name

Let $ m,n$ be positive integers, and let $ F$ be a family of $ m$-element subsets of $ \{1,2,...,n\}$ satisfying $ A\cap B \not \equal{} \emptyset$ for all $ A,B\in F$. Determine the maximum possible number of elements in $ F$.

In a city, there are some buildings. We say the building $A$ is dominant to the building $B$ if the line that connects upside of $A$ to upside of $B$ makes an angle more than $45^{\circ}$ with earth. We want to make a building in a given location. Suppose none of the buildings are dominant to each other. Prove that we can make the building with a height such that again, none of the buildings are dominant to each other. (Suppose the city as a horizontal plain and each building as a perpendicular line to the plain.)

Two persons play a game on a board divided into $3\times 100$ squares. They move in turn: the first places tiles of size $1\times2$ lengthwise (along the long axis of the board), the second, in the perpendicular direction. The loser is the one who cannot make a move. Which of the players can always win (no matter how his opponent plays), and what is the winning strategy?

On each square of an $n\times n$-chessboard, there are two bugs. In a move, each bug moves to a (vertically of horizontally) adjacent square. Bugs from the same square always move to different squares. Determine the maximal number of free squares that can occur after one move. [i](Swiss Mathematical Olympiad 2011, Final round, problem 10)[/i]

Given is $2022\times 2022$ cells table. We can select $4$ cells, such that they make the figure $L$ (rotations, symmetric still count) (left one) and put a ball in each of them, or select $4$ cell which makes up the right figure (rotations, symmetric still count) and get one ball from each of them. For which $k$ is it possible in a given moment to be exactly $k$ points in each of the cells

Initially, the number $1$ and two positive numbers $x$ and $y$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write on the blackboard, in a finite number of moves, the number [list][b]a)[/b] $x^2$; [b]b)[/b] $xy$?[/list]

A Retired Linguist (R.L.) writes in the first move a word consisting of $n$ letters, which are all different. In each move, he determines the maximum $i$, such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Prove that R.L. can make $n!$ moves.

A graph is called $7-chip$ if it obtained by removing at most three edges that have no vertex in common from a complete graph with seven vertices. Consider a complete graph $G$ with $v$ vertices which each edge of its is colored blue or red. Prove that there is either a blue path with $100$ edges or a red $7-chip$.

The vertices of a regular dodecagon are coloured either blue or red. Find the number of all possible colourings which do not contain monochromatic sub-polygons. [i]Vasile Pop[/i]

Given a regular $ 2n$-gon, show that each of its sides and diagonals can be assigned in such a way that the sum of the obtained vectors equals zero.

On a circle, $n \geq 3$ points are marked. Each marked point is coloured red, green or blue. In one step, one can erase two neighbouring marked points of different colours and mark a new point between the locations of the erased points with the third colour. In a final state, all marked points have the same colour which is called the colour of the final state. Find all $n$ for which there exists an initial state of $n$ marked points with one missing colour, from which one can reach a final state of any of the three colours by applying a suitable sequence of steps.

A rectangular $M \times N$ board is divided into $1 \times $ cells. There are also many domino pieces of size $1 \times 2$. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is (a) less than $\frac14 MN$, (b) less than $\frac15 MN$.

The village chatterboxes are exchanging their gossip by phone every day so that any two of them talk to each other exactly once. A certain day, every chatterbox called up at least one of the other chatterboxes. Show that there exist three chatterboxes such that the first called up the second, the second called up the third, and the third called up the first.

A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $\frac{7}{9}$. How many players took part at the tournament?

There is a $50\times 50$ grid board.. Carlos is going to write a number in each box with the following procedure. He first chooses $100$ distinct numbers that we denote $f_1,f_2,f_3,…,f_{50},c_1,c_2,c_3,…,c_{50}$ among which there are exactly $50$ that they are rational. Then he writes in each box ($i,j)$ the number $f_i \cdot c_j$ (the multiplication of $f_i$ by $c_j$). Determine the maximum number of rational numbers that the squares on the board can contain.

[u]Round 1 [/u] [b]p1.[/b] A small pizza costs $\$4$ and has $6$ slices. A large pizza costs $\$9$ and has $14$ slices. If the MMATHS organizers got at least $400$ slices of pizza (having extra is okay) as cheaply as possible, how many large pizzas did they buy? [b]p2.[/b] Rachel flips a fair coin until she gets a tails. What is the probability that she gets an even number of heads before the tails? [b]p3.[/b] Find the unique positive integer $n$ that satisfies $n! \cdot (n + 1)! = (n + 4)!$. [u]Round 2 [/u] [b]p4.[/b] The Portland Malt Shoppe stocks $10$ ice cream flavors and $8$ mix-ins. A milkshake consists of exactly $1$ flavor of ice cream and between $1$ and $3$ mix-ins. (Mix-ins can be repeated, the number of each mix-in matters, and the order of the mix-ins doesn’t matter.) How many different milkshakes can be ordered? [b]p5.[/b] Find the minimum possible value of the expression $(x)^2 + (x + 3)^4 + (x + 4)^4 + (x + 7)^2$, where $x$ is a real number. [b]p6.[/b] Ralph has a cylinder with height $15$ and volume $\frac{960}{\pi}$ . What is the longest distance (staying on the surface) between two points of the cylinder? [u]Round 3 [/u] [b]p7.[/b] If there are exactly $3$ pairs $(x, y)$ satisfying $x^2 + y^2 = 8$ and $x + y = (x - y)^2 + a$, what is the value of $a$? [b]p8.[/b] If $n$ is an integer between $4$ and $1000$, what is the largest possible power of $2$ that $n^4 - 13n^2 + 36$ could be divisible by? (Your answer should be this power of $2$, not just the exponent.) [b]p9.[/b] Find the sum of all positive integers $n \ge 2$ for which the following statement is true: “for any arrangement of $n$ points in three-dimensional space where the points are not all collinear, you can always find one of the points such that the $n - 1$ rays from this point through the other points are all distinct.” [u]Round 4 [/u] [b]p10.[/b] Donald writes the number $12121213131415$ on a piece of paper. How many ways can he rearrange these fourteen digits to make another number where the digit in every place value is different from what was there before? [b]p11.[/b] A question on Joe’s math test asked him to compute $\frac{a}{b} +\frac34$ , where $a$ and $b$ were both integers. Because he didn’t know how to add fractions, he submitted $\frac{a+3}{b+4}$ as his answer. But it turns out that he was right for these particular values of $a$ and $b$! What is the largest possible value that a could have been? [b]p12.[/b] Christopher has a globe with radius $r$ inches. He puts his finger on a point on the equator. He moves his finger $5\pi$ inches North, then $\pi$ inches East, then $5\pi$ inches South, then $2\pi$ inches West. If he ended where he started, what is the largest possible value of $r$? PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2789002p24519497]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The next board is completely covered with dominoes in an arbitrary manner. [img]https://cdn.artofproblemsolving.com/attachments/8/9/b4b791e55091e721c8d6040a65ae6ba788067c.png[/img] a) Prove that we can paint $21$ dominoes in such a way that there are not two dominoes painted forming a $S$-tetramino. b) What is the largest positive integer $k$ for which it is always possible to paint $k$ dominoes (without matter how the board is filled) in such a way that there are not two painted dominoes forming a $S$-tetramine? Clarification: A domino is a $1 \times 2$ or $2 \times 1$ rectangle; the $S$-tetraminos are the figures of the following types: [img]https://cdn.artofproblemsolving.com/attachments/d/f/8480306382d6b87ddb8b2a7ca96c91ee45bc6e.png[/img]

The vertices of a given square are clockwise lettered $A,B,C,D$. On the side $AB$ is situated a point $E$ such that $AE = AB/3$. Starting from an arbitrarily chosen point $P_0$ on segment $AE$ and going clockwise around the perimeter of the square, a series of points $P_0, P_1, P_2, \ldots$ is marked on the perimeter such that $P_iP_{i+1} = AB/3$ for each $i$. It will be clear that when $P_0$ is chosen in $A$ or in $E$, then some $P_i$ will coincide with $P_0$. Does this possibly also happen if $P_0$ is chosen otherwise?

For an integer $n \ge 3$ we consider a circle with $n$ points on it. We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other. For how many values of $n$ with $3 \le n \le 2020$ is it possible to place numbers in a stable way?

Given two integers $n\geq 1$ and $q\geq 2$, let $A=\{(a_1,\ldots ,a_n):a_i\in\{0,\ldots ,q-1\}, i=1,\ldots ,n\}$. If $a=(a_1,\ldots ,a_n)$ and $b=(b_1,\ldots ,b_n)$ are two elements of $A$, let $\delta(a,b)=\#\{i:a_i\neq b_i\}$. Let further $t$ be a non-negative integer and $B$ a non-empty subset of $A$ such that $\delta(a,b)\geq 2t+1$, whenever $a$ and $b$ are distinct elements of $B$. Prove that the two statements below are equivalent: a) For any $a\in A$, there is a unique $b\in B$, such that $\delta (a,b)\leq t$; b) $\displaystyle|B|\cdot \sum_{k=0}^t \binom{n}{k}(q-1)^k=q^n$

There are $n \ge 2$ houses on the northern side of a street. Going from the west to the east, the houses are numbered from 1 to $n$. The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the currnet number plates are swapped exactly once during the day. How many different sequences of number plates are possible at the end of the day?

There are several equal (possibly overlapping) square-shaped napkins on a rectangular table, with sides parallel to the sides of the table. Prove that it is possible to nail some of them to the table in such a way that every napkin is nailed exactly once.

In the class, there are $ 15$ boys and $ 15$ girls. On March $ 8$, some boys made phone calls to some girls to congratulate them on the holiday ( each boy made no more than one call to each girl). It appears that there is a unique way to split the class in $ 15$ pairs (each consisting of a boy and a girl) such that in every pair the boy has phoned the girl. Find the maximal possible number of calls.

There are $36$ cards in a deck arranged in the sequence spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds, etc. Somebody took part of this deck off the top, turned it upside down, and cut this part into the remaining part of the deck (i.e. inserted it between two consecutive cards). Then four cards were taken off the top, then another four, etc. Prove that in any of these sets of four cards, all the cards are of different suits. (A Merkov, Moscow)