[b]p1.[/b] Let $D(n)$ denote the number of positive factors of the integer $n$. For example, $D(6) = 4$ , since the factors of $6$ are $1, 2, 3$ , and $6$ . Note that $D(n) = 2$ if and only if $n$ is a prime number. (a) Describe the set of all solutions to the equation $D(n) = 5$ . (b) Describe the set of all solutions to the equation $D(n) = 6$ . (c) Find the smallest $n$ such that $D(n) = 21$ . [b]p2.[/b] At a party with $n$ married couples present (and no one else), various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening Mr. Jones asked everyone else, including his wife, how many hands he or she had shaken. To his surprise, he got a different answer from each person. Determine the number of hands that Mr. Jones shook that evening, (a) if $n = 2$ . (b) if $n = 3$ . (c) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [b]p3.[/b] Let $n$ be a positive integer. A square is divided into triangles in the following way. A line is drawn from one corner of the square to each of $n$ points along each of the opposite two sides, forming $2n + 2$ nonoverlapping triangles, one of which has a vertex at the opposite corner and the other $2n + 1$ of which have a vertex at the original corner. The figure shows the situation for $n = 2$ . Assume that each of the $2n + 1$ triangles with a vertex in the original corner has area $1$. Determine the area of the square, (a) if $n = 1$ . (b) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [img]https://cdn.artofproblemsolving.com/attachments/1/1/62a54011163cc76cc8d74c73ac9f74420e1b37.png[/img] [b]p4.[/b] Arthur and Betty play a game with the following rules. Initially there are one or more piles of stones, each pile containing one or more stones. A legal move consists either of removing one or more stones from one of the piles, or, if there are at least two piles, combining two piles into one (but not removing any stones). Arthur goes first, and play alternates until a player cannot make a legal move; the player who cannot move loses. (a) Determine who will win the game if initially there are two piles, each with one stone, assuming that both players play optimally. (b) Determine who will win the game if initially there are two piles, each with $n$ stones, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . (c) Determine who will win the game if initially there are $n$ piles, each with one stone, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . [b]p5.[/b] Suppose $x$ and $y$ are real numbers such that $0 < x < y$. Define a sequence$ A_0 , A_1 , A_2, A_3, ...$ by-setting $A_0 = x$ , $A_1 = y$ , and then $A_n= |A_{n-1}| - A_{n-2}$ for each $n \ge 2$ (recall that $|A_{n-1}|$ means the absolute value of $A_{n-1}$ ). (a) Find all possible values for $A_6$ in terms of $x$ and $y$ . (b) Find values of $x$ and $y$ so that $A_{1987} = 1987$ and $A_{1988} = -1988$ (simultaneously). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A traveler arrived to an island where 50 natives lived. All the natives stood in a circle and each announced firstly the age of his left neighbour, then the age of his right neighbour. Each native is either a knight who told both numbers correctly or a knave who increased one of the numbers by 1 and decreased the other by 1 (on his choice). Is it always possible after that to establish which of the natives are knights and which are knaves? [i]Alexandr Gribalko[/i]

[hide=Intro]The answers to each of the ten questions in this section are integers containing only the digits $ 1$ through $ 8$, inclusive. Each answer can be written into the grid on the answer sheet, starting from the cell with the problem number, and continuing across or down until the entire answer has been written. Answers may cross dark lines. If the answers are correctly filled in, it will be uniquely possible to write an integer from $ 1$ to $ 8$ in every cell of the grid, so that each number will appear exactly once in every row, every column, and every marked $2$ by $4$ box. You will get $7$ points for every correctly filled answer, and a $15$ point bonus for filling in every gridcell. It will help to work back and forth between the grid and the problems, although every problem is uniquely solvable on its own. Please write clearly within the boxes. No points will be given for a cell without a number, with multiple numbers, or with illegible handwriting.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/9/b/f4db097a9e3c2602b8608be64f06498bd9d58c.png[/img] [b]1 ACROSS:[/b] Jack puts $ 10$ red marbles, $ 8$ green marbles and 4 blue marbles in a bag. If he takes out $11$ marbles, what is the expected number of green marbles taken out? [b]2 DOWN:[/b] What is the closest integer to $6\sqrt{35}$ ? [b]3 ACROSS: [/b]Alan writes the numbers $ 1$ to $64$ in binary on a piece of paper without leading zeroes. How many more times will he have written the digit $ 1$ than the digit $0$? [b]4 ACROSS:[/b] Integers a and b are chosen such that $-50 < a, b \le 50$. How many ordered pairs $(a, b)$ satisfy the below equation? $$(a + b + 2)(a + 2b + 1) = b$$ [b]5 DOWN: [/b]Zach writes the numbers $ 1$ through $64$ in binary on a piece of paper without leading zeroes. How many times will he have written the two-digit sequence “$10$”? [b]6 ACROSS:[/b] If you are in a car that travels at $60$ miles per hour, $\$1$ is worth $121$ yen, there are $8$ pints in a gallon, your car gets $10$ miles per gallon, a cup of coffee is worth $\$2$, there are 2 cups in a pint, a gallon of gas costs $\$1.50$, 1 mile is about $1.6$ kilometers, and you are going to a coffee shop 32 kilometers away for a gallon of coffee, how much, in yen, will it cost? [b]7 DOWN:[/b] Clive randomly orders the letters of “MIXING THE LETTERS, MAN”. If $\frac{p}{m^nq}$ is the probability that he gets “LMT IS AN EXTREME THING” where p and q are odd integers, and $m$ is a prime number, then what is $m + n$? [b]8 ACROSS:[/b] Joe is playing darts. A dartboard has scores of $10, 7$, and $4$ on it. If Joe can throw $12$ darts, how many possible scores can he end up with? [b]9 ACROSS:[/b] What is the maximum number of bounded regions that $6$ overlapping ellipses can cut the plane into? [b]10 DOWN:[/b] Let $ABC$ be an equilateral triangle, such that $A$ and $B$ both lie on a unit circle with center $O$. What is the maximum distance between $O$ and $C$? Write your answer be in the form $\frac{a\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime, and $a$ and $c$ share no common factor. What is $abc$ ? PS. You had better use hide for answers.

Two people play a game on a $9 \times 9$ board. They move alternately. On each move, the first player draws a cross in an empty cell, and the second player draws a nought in an empty cell. When all $81$ cells are filled, the number $K$ of rows and columns in which there are more crosses and the number $H$ of rows and columns in which there are more noughts are counted. The score for the first player is the difference $B = K- H$. Find a value of $B$ such that the first player can guarantee a score of at least $B$, while the second player can hold the first player's score to at most B, regardless how the opponent plays. (A Kanel)

A square of side $n$ ($n$ natural) is divided into $n^2$ squares of side $1$. Each pair of "horizontal" boundary lines and each pair of "vertical" boundary lines enclose a rectangle (a square is also considered a rectangle). A rectangle has a length and a width; the width is less than or equal to the length. (a) Prove that there are $8$ rectangles of width $n - 1$. (b) Determine the number of rectangles with width $n -k$ ($0\le k \le n -1,k$ integer). (c) Determine a formula for $1^3 + 2^3 +...+ n^3$.

Alice and Bob are play a game in a $(2n)*(2n)$ chess boared.Alice starts from the top right point moving 1 unit in every turn.Bob starts from the down left square and does the same.The goal of Alice is to make a closed shape ending in its start position and cannot reach any point that was reached before by any of players .if a players cannot move the other player keeps moving.what is the maximum are of the shape that the first player can build with every strategy of second player.

We have $10$ boxes of different sizes, each one big enough to contain all the smaller boxes when put side by side. We may nest the boxes however we want (and how deeply we want), as long as we put smaller boxes in larger ones. At the end, all boxes should be directly or indirectly nested in the largest box. How many ways can we nest the boxes?

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

show there exist positive constants $c_1$ and $c_2$ such that for any $n\geq 3$, whenever $T_1$ and $T_2$ are two trees on the set of vertices $X = \{1, 2, ..., n\}$, there exists a function $f : X \to \{-1, +1\}$ for which $$\bigg | \sum_ {x \in P} f (x) \bigg | <c_1 \log n$$ for any path P that is a subgraph of $T_1$ or $T_2$ , but with an upper bound $c_2 \log n / \log \log n$ the statement is no longer true.

[b]a)[/b] Prove that if $G$ is $2$-connected, then it has a cycle with the length at least $\min\{n(G),2\delta(G)\}$. (10 points) [b]b)[/b] Prove that every $2k$-regular graph with $4k+1$ vertices has a hamiltonian cycle. (10 points)

Marine biologists are studying a new species of shellfish whose first generation consists of $100$ shellfish, and their colony reproduces as follows: if a given generation consists of $N$ shellfish (where $5\mid N$ always holds), they divide themselves into $N/5$ groups of $5$ shellfish each. Each group collectively produces $15$ offspring, who form the next generation. Some of the shellfish contain a pearl, but a shellfish can only contain a pearl if none of its direct ancestors contained a pearl. The value of a pearl is determined by the generation of the shellfish containing it: in the $n^{\mathrm{th}}$ generation, its value is $1/3^n$. Find the maximum possible total value of the pearls in the colony. [i]Proposed by: Csongor Beke, Cambridge[/i]

Given the natural $n$. We shall call [i]word [/i] sequence from $n$ letters of the alphabet, and [i]distance [/i] $\rho(A, B)$ between [i]words [/i] $A=a_1a_2\dots a_n$ and $B=b_1b_2\dots b_n$ , the number of digits in which they differ (that is, the number of such $i$, for which $a_i\ne b_i$). We will say that the [i]word [/i] $C$ [i]lies [/i] between words $A$ and $B$ , if $\rho (A,B)=\rho(A,C)+\rho(C,B)$. What is the largest number of [i]words [/i] you can choose so that among any three, there is a [i]word lying[/i] between the other two?

[b]p1.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p2.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from $1$ to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number $3$? The total number of all obtained points is $264$. [b]p2.[/b] There are exactly $n$ students in a high school. Girls send messages to boys. The first girl sent messages to $5$ boys, the second to $7$ boys, the third to $6$ boys, the fourth to $8$ boys, the fifth to $7$ boys, the sixth to $9$ boys, the seventh to $8$, etc. The last girl sent messages to all the boys. Prove that $n$ is divisible by $3$. [b]p4.[/b] In what minimal number of triangles can one cut a $25 \times 12$ rectangle in such a way that one can tile by these triangles a $20 \times 15$ rectangle. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Planet Tarator is a planet in the Yoghurty way galaxy. This planet has a shape of convex $1392$-hedron. On earth we don't have any other information about sides of planet tarator. We have discovered that each side of the planet is a country, and has it's own currency. Each two neighbour countries have their own constant exchange rate, regardless of other exchange rates. Anybody who travels on land and crosses the border must change all his money to the currency of the destination country, and there's no other way to change the money. Incredibly, a person's money may change after crossing some borders and getting back to the point he started, but it's guaranteed that crossing a border and then coming back doesn't change the money. On a research project a group of tourists were chosen and given same amount of money to travel around the Tarator planet and come back to the point they started. They always travel on land and their path is a nonplanar polygon which doesn't intersect itself. What is the maximum number of tourists that may have a pairwise different final amount of money? [b]Note 1:[/b] Tourists spend no money during travel! [b]Note 2:[/b] The only constant of the problem is 1392, the number of the sides. The exchange rates and the way the sides are arranged are unknown. Answer must be a constant number, regardless of the variables. [b]Note 3:[/b] The maximum must be among all possible polyhedras. Time allowed for this problem was 90 minutes.

At a quiz show there are three doors. Behind exactly one of the doors, a prize is hidden. You may ask the quizmaster whether the prize is behind the left-hand door. You may also ask whether the prize is behind the right-hand door. You may ask each of these two questions multiple times, in any order that you like. Each time, the quizmaster will answer ‘yes’ or ‘no’. The quizmaster is allowed to lie at most $10$ times. You have to announce in advance how many questions you will be asking (but which questions you will ask may depend on the answers of the quizmaster). What is the smallest number you can announce, such that you can still determine with absolute certainty the door behind which the prize is hidden?

For a natural number $n$ let $W (n)$ be the number of possibilities, to distribute weights with the masses $1, 2,..., n$ all of them between the two bowls of a beam balance so that they are in balance/ Show that $W (100)$ is really larger than $W (99)$.

Bill Gates and Jeff Bezos are playing a game. Each turn, a coin is flipped, and if Bill and Jeff have $m,n>0$ dollars, respectively, the winner of the coin toss will take $\min{(m,n)}$ from the loser. Given that Bill starts with $20$ dollars and Jeff starts with $21$ dollars, what is the probability that Bill ends up with all of the money? [i]Proposed by Daniel Li[/i]

Let $ n\geq6$ be a even natural number. Prove that any cube can be divided in $ \dfrac{3n(n\minus{}2)}4\plus{}2$ cubes.

( "Sisyphian Labour" ) There are $1001$ steps going up a hill , with rocks on some of them {no more than 1 rock on each step ) . Sisyphus may pick up any rock and raise it one or more steps up to the nearest empty step . Then his opponent Aid rolls a rock (with an empty step directly below it) down one step . There are $500$ rocks, originally located on the first $500$ steps. Sisyphus and Aid move rocks in turn , Sisyphus making the first move . His goal is to place a rock on the top step. Can Aid stop him? ( S . Yeliseyev)

There are three positive integers written on a blackboard every minute. You can pick two written numbers $a$ and $b$ and replace them with $a \cdot b$ and $|a-b|$. Prove that it is always possible to make two of the numbers zero.

Once upon a time there are $n$ pairs of princes and princesses who are in love with each other. One day a witch comes along and turns all the princes into frogs; the frogs can be distinguished by sight but the princesses cannot tell which frog corresponds to which prince. The witch tells the princesses that if any of them kisses the frog that corresponds to the prince very that she loves then that frog will immediately transform back into a prince. If each princess can stand kissing at most $k$ frogs, what is the maximum number of princes they can be sure to save? (The princesses may take turns kissing in any order, communicate with each other and vary their strategy for future kisses depending on information gained from past kisses.)

Let $n$ be an integer greater than $1$ and let $S$ be a finite set containing more than $n+1$ elements.Consider the collection of all sets $A$ of subsets of $S$ satisfying the following two conditions : [b](a)[/b] Each member of $A$ contains at least $n$ elements of $S$. [b](b)[/b] Each element of $S$ is contained in at least $n$ members of $A$. Determine $\max_A \min_B |B|$ , as $B$ runs through all subsets of $A$ whose members cover $S$ , and $A$ runs through the above collection.

Initially there is a checker on every square of a $1\times n$ board. The first move consists of moving a checker to an adjacent square thus creating a stack of two checkers. Then each time when making a move, one can choose a stack and move it in either direction as many squares on the board as there are checkers in the stack. If after the move the stack lands on a non-empty square, it is placed on top of the stack which is already there. Prove that it is possible to stack all the checkers on one square in $n - 1$ moves. (A Shapovalov)

How many ways are there to place the numbers $2, 3, . . . , 10$ in a $3 \times 3$ grid, such that any two numbers that share an edge are mutually prime?

In a room with $9$ students, there are $n$ clubs with $4$ participants in each club. For any pairs of clubs no more than $2$ students belong to both clubs. Prove that $n \le 18$ [i]Proposed by Manuel Aguilera, Valle[/i]