Removing a unit square from a $2\times 2$ square we get a piece called [i]L-tromino.[/i] From the fourth line of a $7 \times 7$ cheesboard some unit squares have been removed. The resulting chessboard is cut in L-trominos. Determine the number and location of the removed squares.

In a game of rock-paper-scissors with $n$ people, the following rules are used to determine a champion: (a) In a round, each person who has not been eliminated randomly chooses one of rock, paper, or scissors to play. (b) If at least one person plays rock, at least one person plays paper, and at least one person plays scissors, then the round is declared a tie and no one is eliminated. If everyone makes the same move, then the round is also declared a tie. (c) If exactly two moves are represented, then everyone who made the losing move is eliminated from playing in all further rounds (for example, in a game with $8$ people, if $5$ people play rock and $3$ people play scissors, then the $3$ who played scissors are eliminated). (d) The rounds continue until only one person has not been eliminated. That person is declared the champion and the game ends. If a game begins with $4$ people, what is the expected value of the number of rounds required for a champion to be determined? [i]In the game of rock-paper-scissors, two players each choose one of rock, paper, or scissors to play. Rock beats scissors, scissors beats paper, and paper beats rock. If the players play the same thing, the match is considered a draw.[/i]

[b]p13.[/b] Emma has the five letters: $A, B, C, D, E$. How many ways can she rearrange the letters into words? Note that the order of words matter, ie $ABC DE$ and $DE ABC$ are different. [b]p14.[/b] Seven students are doing a holiday gift exchange. Each student writes their name on a slip of paper and places it into a hat. Then, each student draws a name from the hat to determine who they will buy a gift for. What is the probability that no student draws himself/herself? [b]p15.[/b] We model a fidget spinner as shown below (include diagram) with a series of arcs on circles of radii $1$. What is the area swept out by the fidget spinner as it’s turned $60^o$ ? [img]https://cdn.artofproblemsolving.com/attachments/9/8/db27ffce2af68d27eee5903c9f09a36c2a6edf.png[/img] [b]p16.[/b] Let $a,b,c$ be the sides of a triangle such that $gcd(a, b) = 3528$, $gcd(b, c) = 1008$, $gcd(a, c) = 504$. Find the value of $a * b * c$. Write your answer as a prime factorization. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Pupils of a school are divided into $112$ groups, of $11$ members each. Any two groups have exactly one common pupil. Prove that: a) there is a pupil that belongs to at least $12$ groups. b) there is a pupil that belongs to all the groups.

A set consisting of $n$ points of a plane is called a [i]bosonti $n$-point[/i] if any three of its points are located in vertices of an isosceles triangle. Find all positive integers $n$ for which there exists a bosonti $n$-point.

Each student in a classroom has $0,1,2,3,4,5$ or $6$ pieces of candy. At each step the teacher chooses some of the students, and gives one piece of candy to each of them and also to any other student in the classroom who is friends with at least one of these students. A student who receives the seventh piece eats all $7$ pieces. Assume that for every pair of students in the classroom, there is at least one student who is friend swith exactly one of them. Show that no matter how many pieces each student has at the beginning, the teacher can make them to have any desired numbers of pieces after finitely many steps.

[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a school, there are 2021 students, each having exactly $k$ friends. There aren't three students such that all three are friends with each other. What is the maximum possible value of $k$?

An invisible tank is on a $100 \times 100 $ table. A cannon can fire at any $60$ cells of the board after that the tank will move to one of the adjacent cells (by side). Then the progress is repeated. Can the cannon grantee to shoot the tank?

One positive integer is written in each $1 \times 1$ square of the $m \times n$ board. The following operations are allowed : (1) In an arbitrarily selected row of the board, all numbers should be reduced by $1$. (2) In an arbitrarily selected column of the board, double all the numbers. Is it always possible, after a final number of steps, for all the numbers written on the board to be equal to $-1$? (Explain the answer.)

Consider a rectangular table $2 \times n.$ Let every cell be dyed either by black or white color in a way that no $2\times 2$ square is completely black. Denote $P_n$ the number of such colorings. Prove that the number $P_{1989}$ is divisible by three and find the greatest power of three that divides them.

Let $ 0 < a_1 < a_2 < a_3 < ... < a_{10000} < 20000$ be integers such that $ gcd(a_i,a_j) < a_i, \forall i < j$ ; is $ 500 < a_1$ [i](always)[/i] true ? [i](own)[/i] :lol:

Justin is being served two different types of chips, A-chips, and B-chips. If there are $3$ B-chips and $5$ A-chips, and if Justin randomly grabs $3$ chips, what is the probability that none of them are A-chips?

For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically? Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.

There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.) [center][img]http://i.imgur.com/4ggOa8m.png[/img][/center] [list=a] [*]Can the puzzle be solved for $n=5$? [*]Can the puzzle be solved for $n=2014$? [/list] In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.

Given a table in a form of the regular $1000$-gon with sidelength $1$. A Beetle initially is in one of its vertices. All $1000$ vertices are numbered in some order by numbers $1$, $2$, $\ldots$, $1000$ so that initially the Beetle is in the vertex $1$. The Beetle can move only along the edges of $1000$-gon and only clockwise. He starts to move from vertex $1$ and he is moving without stops until he reaches vertex $2$ where he has a stop. Then he continues his journey clockwise from vertex $2$ until he reaches the vertex $3$ where he has a stop, and so on. The Beetle finishes his journey at vertex $1000$. Find the number of ways to enumerate all vertices so that the total length of the Beetle's journey is equal to $2017$.

Call arrangement of $m$ number on the circle [b]$m$-negative[/b], if all numbers are equal to $-1$. On the first step Andrew chooses one number on circle and multiplies it by $-1$. All other steps are similar: instead of the next number(clockwise) he writes its product with the number, written on the previous step. Prove that if $n$-negative arrangement in $k$ steps becomes $n$-negative again, then $(2^n - 1)$-negative after $(2^k - 1)$ steps becomes $(2^n - 1)$-negative again.

The following sequence of letters is written on a board: \[ \text{TNOTNOTNO...TNOTN} \] where the sequence repeats 2024 times. At each step, one of the following operations can be performed: 1. Take two different adjacent letters and replace them with two copies of the missing letter. 2. Take three consecutive identical letters and remove them. After a certain number of steps, only two identical letters remain. Determine which letter it is possible to reach.

In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).

Let $n$ points $A_1, A_2, \ldots, A_n$, ($n>2$), be considered in the space, where no four points are coplanar. Each pair of points $A_i, A_j$ are connected by an edge. Find the maximal value of $n$ for which we can paint all edges by two colors – blue and red such that the following three conditions hold: [b]I.[/b] Each edge is painted by exactly one color. [b]II.[/b] For $i = 1, 2, \ldots, n$, the number of blue edges with one end $A_i$ does not exceed 4. [b]III.[/b] For every red edge $A_iA_j$, we can find at least one point $A_k$ ($k \neq i, j$) such that the edges $A_iA_k$ and $A_jA_k$ are blue.

Find the number of ways to arrange the letters in $LE X I NGTON$ such that the string $LE X$ does not appear.

Compute the number of ways there are to completely fill a $3\times 15$ rectangle with non-overlapping $1\times 3$ rectangles

We deal $n-1$ cards in some way to $n$ people sitting around a table. From then on, in one move a person with at least $2$ cards gives one card to each of his/her neighbors. Prove that eventually a state will be reached where everyone has at most one card.

The five sides and five diagonals of a regular pentagon are drawn on a piece of paper. Two people play a game, in which they take turns to colour one of these ten line segments. The first player colours line segments blue, while the second player colours line segments red. A player cannot colour a line segment that has already been coloured. A player wins if they are the first to create a triangle in their own colour, whose three vertices are also vertices of the regular pentagon. The game is declared a draw if all ten line segments have been coloured without a player winning. Determine whether the first player, the second player, or neither player can force a win.

Find the number of four-element subsets of $\{1,2,3,4,\dots, 20\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\{3,5,13,19\}$ and $\{6,10,20,18\}$ are two such subsets.