[u]Set 1[/u] [b]p1.[/b] Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters? [b]p2.[/b] Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get? [b]p3.[/b] Twain is trying to crack a $4$-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules? PS. You should use hide for answers.

$ (a)$ A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present? $ (b)$ If, in addition, the group contains three mutual acquaintances, what is the maximum possible number of people?

Initially given $31$ tuplets $$(1,0,0,\dots,0),(0,1,0,\dots,0),\dots, (0,0,0,\dots,1)$$ were written on the blackboard. At every move we choose two written $31$ tuplets as $(a_1,a_2,a_3,\dots, a_{31})$ and $(b_1,b_2,b_3,\dots,b_{31})$, then write the $31$ tuplet $(a_1+b_1,a_2+b_2,a_3+b_3,\dots, a_{31}+b_{31})$ to the blackboard too. Find the least possible value of the moves such that one can write the $31$ tuplets $$(0,1,1,\dots,1),(1,0,1,\dots,1),\dots, (1,1,1,\dots,0)$$ to the blackboard by using those moves.

Alice has $100$ balls and $10$ buckets. She takes each ball and puts it in a bucket that she chooses at random. After she is done, let $b_i$ be the number of balls in the $i$th bucket, for $1\le i \le 10$. Compute the expected value of $\Sigma_{i=1}^{10}b_i^2$

[b]p1.[/b] Sometimes one finds in an old park a tetrahedral pile of cannon balls, that is, a pile each layer of which is a tightly packed triangular layer of balls. A. How many cannon balls are in a tetrahedral pile of cannon balls of $N$ layers? B. How high is a tetrahedral pile of cannon balls of $N$ layers? (Assume each cannon ball is a sphere of radius $R$.) [b]p2.[/b] A prime is an integer greater than $1$ whose only positive integer divisors are itself and $1$. A. Find a triple of primes $(p, q, r)$ such that $p = q + 2$ and $q = r + 2$ . B. Prove that there is only one triple $(p, q, r)$ of primes such that $p = q + 2$ and $q = r + 2$ . [b]p3.[/b] The function $g$ is defined recursively on the positive integers by $g(1) =1$, and for $n>1$ , $g(n)= 1+g(n-g(n-1))$ . A. Find $g(1)$ , $g(2)$ , $g(3)$ and $g(4)$ . B. Describe the pattern formed by the entire sequence $g(1) , g(2 ), g(3), ...$ C. Prove your answer to Part B. [b]p4.[/b] Let $x$ , $y$ and $z$ be real numbers such that $x + y + z = 1$ and $xyz = 3$ . A. Prove that none of $x$ , $y$ , nor $z$ can equal $1$. B. Determine all values of $x$ that can occur in a simultaneous solution to these two equations (where $x , y , z$ are real numbers). [b]p5.[/b] A round robin tournament was played among thirteen teams. Each team played every other team exactly once. At the conclusion of the tournament, it happened that each team had won six games and lost six games. A. How many games were played in this tournament? B. Define a [i]circular triangle[/i] in a round robin tournament to be a set of three different teams in which none of the three teams beat both of the other two teams. How many circular triangles are there in this tournament? C. Prove your answer to Part B. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a table there are $2013$ cards that have written, each one, a different integer number, from $1$ to $2013$; all the cards face down (you can't see what number they are). It is allowed to select any set of cards and ask if the average of the numbers written on those cards is integer. The answer will be true. a) Find all the numbers that can be determined with certainty by several of these questions. b) We want to divide the cards into groups such that the content of each group is known even though the individual value of each card in the group is not known. (For example, finding a group of $3$ cards that contains $1, 2$, and $3$, without knowing what number each card has.) What is the maximum number of groups that can be obtained?

A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)

A set of $n$ lights, numbered $1$ to $n$, are initially off. At every moment, it is possible to perform one of the following operations: $\bullet$ change the state of lamp $1$, $\bullet$ change the state of lamp $2$, as long as lamp $1$ is on, $\bullet$ change the state of lamp $k > 2$, as long as lamp $k - 1$ is on and all lamps $1, . . . , k - 2$ are off. It shows that it is possible, after a certain number of operations, to have only the lamp left on.

Each vertex of a cube is assigned $1$ or $-1$. Each face is assigned the product of the four numbers at its vertices. Determine all possible values that can be obtained as the sum of all the $14$ assigned numbers.

Boy A and $2016$ flags are on a circumference whose length is $1$ of a circle. He wants to get all flags by moving on the circumference. He can get all flags by moving distance $l$ regardless of the positions of boy A and flags. Find the possible minimum value as $l$ like this. Note that boy A doesn’t have to return to the starting point to leave gotten flags.

Let $M\subset \{1, 2, 3, . . . , 2007\}$ a set with the following property: Among every three numbers one can always choose two from $M$ such that one is divisible by the other. How many numbers can $M$ contain at most?

Let $k>1$ be a positive integer. On a $\text{k} \times \text{k}$ square grid, Tom and Jerry are on opposite corners, with Tom at the top right corner. Both can move to an adjacent square every move, where two squares are adjacent if they share a side. Tom and Jerry alternate moves, with Jerry going first. Tom [i]catches[/i] Jerry if they are on the same square. We aim to answer to the following question: What is the smallest number of moves that Tom needs to guarantee catching Jerry? (a) Without proof, find the answer in the cases of $k=2,3,4$, and (correctly) guess what the answer is in terms of $k$. We'll refer to this answer as $A(k)$. (b) Find a strategy that Jerry can use to guarantee that Tom takes at least $A(k)$ moves to catch Jerry. Now, you will find a strategy for Tom to catch Jerry in at most $A(k)$ moves, no matter what Jerry does. (c) Find, with proof, a working strategy for $k=5$. (d) Find, with proof, a working strategy for all $k \geq 2$.

On a rectangular board with $m \times n$ squares ($m, n \ge 3$) there are dominoes ($2 \times 1$ or $1\times 2$ tiles), which do not overlap and do not extend beyond the board. Every domino covers exactly two squares of the board. Assume that the dominos cover the has the property that no more dominos can be added to the board and that the four corner spaces of the board are not all empty. Prove that at least $2/3$ of the squares of the board are covered with dominos.

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

A large golden square land lot of dimension $100 \times 100$ m was subdivided into $100$ square lots, each measured $10\times10$ m. A king of landfill had his men dump wastes onto some of the lots. There was a practice that if a particular lot was not dumped and twoof its adjacents had waste materials, then the lot would be filled with wastes the next day by the people. One day if all the lotswere filled with wastes, the king would claim his ownership ofthe whole land lot. At least how many lots should have the kind had his men dump wastes onto? Vu Ha Van, Mathematics Faculty, Yale University, USA.

Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids". Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".

Given is a convex polygon with $n\geq 5$ sides. Prove that there exist at most $\displaystyle \frac{n(2n-5)}3$ triangles of area 1 with the vertices among the vertices of the polygon.

How many paths are there from $A$ to $B$ that consist of $5$ horizontal segments and $5$ vertical segments of length $1$ each? (see figure) [img]https://cdn.artofproblemsolving.com/attachments/4/2/5b476ca2a232fc67fb2e2f6bb06111cab60692.png[/img]

Let $n$ be a positive integer, the players A and B play the following game: we have $n$ balls with the numbers of $1, 2, 3, 4,...., n$ this balls will be in two boxes with the symbols $\prod$ and $\sum$. In your turn, the player can choose one ball and the player will put this ball in some box, in the final all the balls of the box $\prod$ are multiplied and we will get a number $P$, after this all the balls of the box $\sum$ are added up and we will get a number $Q$(if the box $\prod$ is empty $P = 1$, if the box $\sum$ is empty $Q = 0$). The player(s) play alternately, player A starts, if $P + Q$ is even player A wins, otherwise player B wins. a)If $n= 6$, which player has the winning strategy??? b)If $n = 2012$, which player has the winning strategy???

Let $(v_1, ..., v_{2^n})$ be the vertices of an $n$-dimensional hypercube. Label each vertex $v_i$ with a real number $x_i$. Label each edge of the hypercube with the product of labels of the two vertices it connects. Let $S$ be the sum of the labels of all the edges. Over all possible labelings, find the minimum possible value of $\frac{S}{x^2_1+ x^2_2+ ...+ x^2_n}$ in terms of $ n$. Note: an $n$ dimensional hypercube is a graph on $2^n$ vertices labeled labeled with the binary strings of length $n$, where two vertices have an edge between them if and only if their labels differ in exactly one place. For instance, the vertices $100$ and $101$ on the $3$ dimensional hypercube are connected, but the vertices $100$ and $111$ are not.

There are $n$ $(n \ge 3)$ players in a table tennis tournament, in which any two players have a match. Player $A$ is called not out-performed by player $B$, if at least one of player $A$'s losers is not a $B$'s loser. Determine, with proof, all possible values of $n$, such that the following case could happen: after finishing all the matches, every player is not out-performed by any other player.

We drew several straight lines on the plane and marked all of them intersection points. How many lines could be drawn? if one point is marked on one of the drawn lines, on the other - three, and on the third - five? Find all possible options and prove that there are no others.

$n$ people numbered from $1$ to $n$ are arranged in a row. An [i]acceptable movement[/i] consists of each person changing at most once its place with another or remains in its place. For example $\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline initial position & 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 & n \\ \hline final position & 2 & 1 & 3 & 6 & 5 & 4 & ... & n & n-1 & n-2 \\ \hline \end{tabular}$ is an a[i]cceptable movement[/i]. Is it possible that starting from the position $\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 & n \\ \hline \end{tabular}$ to reach to $\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & ... & n-2 & n-1 \\ \hline \end{tabular}$ through two [i]acceptable movements[/i]?

A right hexagon with side length $n$ is divided into tiles of three types, which are shown in the image, which are rhombuses with side length $1$ each and the acute angle $60$. In one move you can choose three tiles, arranged as shown in the image on the left, and rearrange them, as shown in the image on the right [img]https://iili.io/dxEvyqN.jpg[/img] Moves are made until it is impossible to make a move. a) Prove that for the fixed initial arrangement of tiles the same amount of moves would be made independent of the moves. b) For each positive integer $n$ find the maximum number of moves among all possible initial arrangements [i]M. Zorka[/i]

Several strips and a circle of radius $1$ are drawn on the plane. The sum of the widths of the strips is $100$. Prove that one can translate each strip parallel to itself so that together they cover the circle. (M Smurov )