Kelvin and Quinn are collecting trading cards; there are $6$ distinct cards that could appear in a pack. Each pack contains exactly one card, and each card is equally likely. Kelvin buys packs until he has at least one copy of every card, and then he stops buying packs. If Quinn is missing exactly one card, the probability that Kelvin has at least two copies of the card Quinn is missing is expressible as $\tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Determine $m+n$.

Let $S$ be the set of colorings of a $100 \times 100$ grid where each square is colored black or white and no $2\times2$ subgrid is colored like a chessboard. A random such coloring is chosen: what is the probability there is a path of black squares going from the top row to the bottom row where any two consecutive squares in the path are adjacent? [i]Proposed by Evan Chang (squareman), USA [/i]

Given integer $n\geq 2$, two invisible [color=#eee]rabbits[/color] (rabbits) discussed their strategy and was sent to two points $A, B$ with distance $n$ units on a plane, respectively. However, they do not know whether they are on the same or different side of the plane (when facing each other, the might view the left/right direction as the same or different). They both can see points $A,B$, and they need to hop to each other's starting point. Each hop would measure $1$ unit in distance, and they would jump and land at the same time for each round. However, if at any time they landed no more than $1$ unit away, they'll both turn into deer. Find the minimum number of round they need to reach their destiny while ensuring they won't turn into deer. [i]Proposed by redshrimp[/i]

One needs to ffll the cells of an $n\times n$ table ($n > 1$) with distinct integers from $1$ to $n^2$ so that every two consecutive integers are placed in cells that share a side, while every two integers with the same remainder if divided by $n$ are placed in distinct rows and distinct columns. For which $n$ is this possible? (Alexandr Gribalko)

What is the largest number of obtuse triangles that can be composed of $16$ different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?

$n \geq 5$ football teams participate in a round-robin tournament. For every game played, the winner receives 3 points, the loser receives 0 points, and in the event of a draw, both teams receive 1 point. The third-from-bottom team has fewer points than all the teams ranked before it, and more points than the last 2 teams; it won more games than all the teams before it, but fewer games than the 2 teams behind it. Find the smallest possible $n$.

There are 100 persons in a hall, everyone knowing at least 67 of the others. Prove that there always exist four of them who know each other

On a 8 x 8 board we place $n$ dominoes, each covering two adjacent squares, so that no more dominoes can be placed on the remaining squares. What is the smallest value of $n$ for which the above statement is true?

There are $n$ boys $B_1, B_2, ... , B_n$ and $n$ girls $G_1, G_2, ... , G_n$. Each boy ranks the girls in order of preference, and each girl ranks the boys in order of preference. Show that we can arrange the boys and girls into n pairs so that we cannot find a boy and a girl who prefer each other to their partners. For example if $(B_1, G_3)$ and $(B_4, G_7)$ are two of the pairs, then it must not be the case that $B_4$ prefers $G_3$ to $G_7$ and $G_3$ prefers $B_4$ to $B_1$.

Let $n$ be a fixed positive integer. Oriol has $n$ cards, each of them with a $0$ written on one side and $1$ on the other. We place these cards in line, some face up and some face down (possibly all on the same side). We begin the following process consisting of $n$ steps: 1) At the first step, Oriol flips the first card 2) At the second step, Oriol flips the first card and second card . . . n) At the last step Oriol flips all the cards Let $s_0, s_1, s_2, \dots, s_n$ be the sum of the numbers seen in the cards at the beggining, after the first step, after the second step, $\dots$ after the last step, respectively. a) Find the greatest integer $k$ such that, no matter the initial card configuration, there exists at least $k$ distinct numbers between $s_0, s_1, \dots, s_n$. b) Find all positive integers $m$ such that, for each initial card configuration, there exists an index $r$ such that $s_r = m$. [i]Proposed by Dorlir Ahmeti[/i]

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a [i]clique[/i] if each two of them are friends. (In particular, any group of fewer than two competitiors is a clique.) The number of members of a clique is called its [i]size[/i]. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room. [i]Author: Vasily Astakhov, Russia[/i]

Consider a finite sequence $a_1, a_2,...,a_n$ whose terms are natural numbers at most equal to $n$. Determine the maximum number of terms of such a sequence, if you know that every two of its neighboring terms are different and at the same time there is no quartet of terms in it such that $a_p = a_r \ne a_q = a_s$ for $p < q < r < s$.

Joining $15^3 = 3375$ cubes of $1$ cm$^3$, bodies with a volume of $3375$ cm$^3$ can be built. Indicate how two bodies $A$ and $B$ are constructed with $3375$ cubes each and such that the lateral surface of $B$ is $10$ times the lateral surface of $A$.

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

On a social network, no user has more than ten friends ( the state "friendship" is symmetrical). The network is connected: if, upon learning interesting news a user starts sending it to its friends, and these friends to their own friends and so on, then at the end, all users hear about the news. Prove that the network administration can divide users into groups so that the following conditions are met: [list] [*] each user is in exactly one group [*] each group is connected in the above sense [*] one of the groups contains from $1$ to $100$ members and the remaining from $100$ to $900$. [/list]

(a) A radio lamp has a $7$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $7$ holes. Is it possible to number the contacts and the holes so that for any insertion at least one contact would match the hole with the same number? (b) A radio lamp has a $20$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $20$ holes. Let the contacts in the plug and the socket be already numbered. Is it always possible to insert the plug so that none of the contacts matches its socket?

Find all positive integers $n \geq 3$ for which there exists a set $S$ which consists of rational numbers such that the following two conditions hold: 1) any rational number can be represented as the sum of at most $n$ elements of $S$ 2) there exists a rational number, which can not be represented as the sum of at most $n-1$ elements of $S$ (in the sum some elements can repeat) [i]M. Shutro, M. Zorka[/i]

In the Cartesian plane is given a polygon $P$ whose vertices have integer coordinates and with sides parallel to the coordinate axes. Show that if the length of each edge of $P$ is an odd integer, then the surface of P cannot be partitioned into $2\times 1$ rectangles.

There are $28$ students who have to be separated into two groups such that the number of students in each group is a multiple of $4$. The number of ways to split them into the groups can be written as $$\sum_{k \ge 0} 2^k a_k = a_0 +2a_1 +4a_2 +...$$ where each $a_i$ is either $0$ or $1$. Find the value of $$\sum_{k \ge 0} ka_k = 0+ a_1 +2a_2 +3a3_ +....$$

A grasshopper is jumping about in a grid. From the point with coordinates $(a, b)$ it can jump to either $(a + 1, b),(a + 2, b),(a + 1, b + 1),(a, b + 2)$ or $(a, b + 1)$. In how many ways can it reach the line $x + y = 2014?$ Where the grasshopper starts in $(0, 0)$.

[b]2.[/b] Place 32 white and 32 black chessmen on the chessboard. Two chessmen of different colours will be said to form a "related pair" if they are placed either in the same row or in the same column. Determine the maximum and minimum number of related pairs (over all possible arrangements of the 64 chessmen considered. [b](C. 2)[/b]

Show that the set $\{1,2,....,2^n\}$ can be partitioned in two classes, none of which contains an arithmetic progression of length $2n$.

Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns. Determine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, with the following condition holds: in any $k$[i]-group[/i] from the colored $N$ unit squares, there are two squares with the same color, and there are also two squares with different colors.

[b]p1.[/b] The case of Mr. Brown, Mr. Potter, and Mr. Smith is investigated. One of them has committed a crime. Everyone of them made two statements. Mr. Brown: I have not done it. Mr. Potter has not done it. Mr. Potter: Mr. Brown has not done it. Mr. Smith has done it. Mr. Smith: I have not done it. Mr. Brown has done it. It is known that one of them told the truth both times, one lied both times, and one told the truth one time and lied one time. Who has committed the crime? [b]p2.[/b] Is it possible to draw in a plane $1000001$ circles of the radius $1$ such that every circle touches exactly three other circles? [b]p3.[/b] Consider a circle of radius $R$ centered at the origin. A particle is “launched” from the $x$-axis at a distance, $d$, from the origin with $0 < d < R$, and at an angle, $\alpha$, with the $x$-axis. The particle is reflected from the boundary of the circle so that the [b]angle of incidence[/b] equals the [b]angle of reflection[/b]. Determine the angle $\alpha$ so that the path of the particle contacts the circle’s interior at exactly eight points. Please note that $\alpha$ should be determined in terms of the qunatities $R$ and $d$. [img]https://cdn.artofproblemsolving.com/attachments/e/3/b8ef9bb8d1b54c263bf2b68d3de60be5b41ad0.png[/img] [b]p4.[/b] Is it possible to find four different real numbers such that the cube of every number equals the square of the sum of the three others? [b]p5. [/b]The Fibonacci sequence $(1, 2, 3, 5, 8, 13, 21, . . .)$ is defined by the following formula: $f_n = f_{n-2} + f_{n-1}$, where $f_1 = 1$, $f_2 = 2$. Prove that any positive integer can be represented as a sum of different members of the Fibonacci sequence. [b]p6.[/b] $10,000$ points are arbitrary chosen inside a square of area $1$ m$^2$ . Does there exist a broken line connecting all these points, the length of which is less than $201$ m$^2? PS. You should use hide for answers.

A function $f:\{1,2, \ldots, n\} \to \{1, \ldots, m\}$ is [i]multiplication-preserving[/i] if $f(i)f(j) = f(ij)$ for all $1 \le i \le j \le ij \le n$, and [i]injective[/i] if $f(i) = f(j)$ only when $i = j$. For $n = 9, m = 88$, the number of injective, multiplication-preserving functions is $N$. Find the sum of the prime factors of $N$, including multiplicity. (For example, if $N = 12$, the answer would be $2 + 2 + 3 = 7$.)