Two players, $A$ and $B$, play a game on a board which is a rhombus of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles, as shown in the diagram for $n=4$. $A$ uses a red token and $B$ uses a blue token, which are initially placed in cells containing opposite corners of the board (the $60^{\circ}$ ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started. If $A$ starts the game, determine which player has a winning strategy.

[b]p1.[/b] Compute $x$ such that $2009^{2010} \equiv x$ (mod $2011$) and $0 \le x < 2011$. [b]p2.[/b] Compute the number of "words" that can be formed by rearranging the letters of the word "syzygy" so that the y's are evenly spaced. (The $y$'s are evenly spaced if the number of letters (possibly zero) between the first $y$ and the second $y$ is the same as the number of letters between the second $y$ and the third $y$.) [b]p3.[/b] Let $A$ and $B$ be subsets of the integers, and let $A + B$ be the set containing all sums of the form $a + b$, where $a$ is an element of $A$, and $b$ is an element of $B$. For example, if $A = \{0, 4, 5\}$ and $B =\{-3,-1, 2, 6\}$, then $A + B = \{-3,-1, 1, 2, 3, 4, 6, 7, 10, 11\}$. If $A$ has $1955$ elements and $B$ has $1891$ elements, compute the smallest possible number of elements in $A + B$. [b]p4.[/b] Compute the sum of all integers of the form $p^n$ where $p$ is a prime, $n \ge 3$, and $p^n \le 1000$. [b]p5.[/b] In a season of interhouse athletics at Caltech, each of the eight houses plays each other house in a particular sport. Suppose one of the houses has a $1/3$ chance of beating each other house. If the results of the games are independent, compute the probability that they win at least three games in a row. [b]p6.[/b] A positive integer $n$ is special if there are exactly $2010$ positive integers smaller than $n$ and relatively prime to $n$. Compute the sum of all special numbers. [b]p7.[/b] Eight friends are playing informal games of ultimate frisbee. For each game, they split themselves up into two teams of four. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Determine the smallest number of games they need to play in order to achieve this. [b]p8.[/b] Compute the number of ways to choose five nonnegative integers $a, b, c, d$, and $e$, such that $a + b + c + d + e = 20$. [b]p9.[/b] Is $23$ a square mod $41$? Is $15$ a square mod $41$? [b]p10.[/b] Let $\phi (n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Compute $ \sum_{d|15015} \phi (d)$. [b]p11.[/b] Compute the largest possible volume of an regular tetrahedron contained in a cube with volume $1$. [b]p12.[/b] Compute the number of ways to cover a $4 \times 4$ grid with dominoes. [b]p13.[/b] A collection of points is called mutually equidistant if the distance between any two of them is the same. For example, three mutually equidistant points form an equilateral triangle in the plane, and four mutually equidistant points form a regular tetrahedron in three-dimensional space. Let $A$, $B$, $C$, $D$, and $E$ be five mutually equidistant points in four-dimensional space. Let $P$ be a point such that $AP = BP = CP = DP = EP = 1$. Compute the side length $AB$. [b]p14. [/b]Ten turtles live in a pond shaped like a $10$-gon. Because it's a sunny day, all the turtles are sitting in the sun, one at each vertex of the pond. David decides he wants to scare all the turtles back into the pond. When he startles a turtle, it dives into the pond. Moreover, any turtles on the two neighbouring vertices also dive into the pond. However, if the vertex opposite the startled turtle is empty, then a turtle crawls out of the pond and sits at that vertex. Compute the minimum number of times David needs to startle a turtle so that, by the end, all but one of the turtles are in the pond. [b]p15.[/b] The game hexapawn is played on a $3 \times 3$ chessboard. Each player starts with three pawns on the row nearest him or her. The players take turns moving their pawns. Like in chess, on a player's turn he or she can either $\bullet$ move a pawn forward one space if that square is empty, or $\bullet$ capture an opponent's pawn by moving his or her own pawn diagonally forward one space into the opponent's pawn's square. A player wins when either $\bullet$ he or she moves a pawn into the last row, or $\bullet$ his or her opponent has no legal moves. Eve and Fred are going to play hexapawn. However, they're not very good at it. Each turn, they will pick a legal move at random with equal probability, with one exception: If some move will immediately win the game (by either of the two winning conditions), then he or she will make that move, even if other moves are available. If Eve moves first, compute the probability that she will win. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Petya and Kolya play the following game: they take turns changing one of the coefficients $a$ or $b$ of the quadratic trinomial $f = x^2 + ax + b$: Petya is on $1$, Kolya is on $1$ or $3$. Kolya wins if after the move of one of the players a trinomial is obtained that has whole roots. Is it true that Kolya can win for any initial integer odds $a$ and $b$ regardless of Petya's game? [hide=original wording]Петя и Коля играют в следующую игру: они по очереди изменяют один из коэффициентов a или b квадратного трехчлена f = x^2 + ax + b: Петя на 1, Коля- на 1 или на 3. Коля выигрывает, если после хода одного из игроков получается трехчлен, имеющий целые корни. Верно ли, что Коля может выигратьпр и любых начальных целых коэффициентах a и b независимо от игры Пети?[/hide]

Alice and Bob play a game on a Cartesian Coordinate Plane. At the beginning, Alice chooses a lattice point $ \left(x_{0}, y_{0}\right) $ and places a pudding. Then they plays by turns (B goes first) according to the rules a. If $ A $ places a pudding on $ \left(x,y\right) $ in the last round, then $ B $ can only place a pudding on one of $ \left(x+2, y+1\right), \left(x+2, y-1\right), \left(x-2, y+1\right), \left(x-2, y-1\right) $ b. If $ B $ places a pudding on $ \left(x,y\right) $ in the last round, then $ A $ can only place a pudding on one of $ \left(x+1, y+2\right), \left(x+1, y-2\right), \left(x-1, y+2\right), \left(x-1, y-2\right) $ Furthermore, if there is already a pudding on $ \left(a,b\right) $, then no one can place a pudding on $ \left(c,d\right) $ where $ c \equiv a \pmod{n}, d \equiv b \pmod{n} $. 1. Who has a winning strategy when $ n = 2018 $ 1. Who has a winning strategy when $ n = 2019 $

Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. A move is for a grasshopper to select a marked point and jump over it to the equidistant point the other side. This point must lie in the interval for the move to be allowed, but it does not have to be marked. What is the smallest $n$ such that if each grasshopper makes $n$ moves or less, then they end up with no marked points between them?

All positive integers from \( 1 \) to \( 2025 \) are written on a board. Mykhailo and Oleksii play the following game. They take turns, starting with Mykhailo, erasing one of the numbers written on the board. The game ends when exactly two numbers remain on the board. If their sum is a perfect square of an integer, Mykhailo wins; otherwise, Oleksii wins. Who wins if both players play optimally? [i]Proposed by Fedir Yudin[/i]

Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with ($a,b)$, produces new card with $(a+1,b+1)$, the second, having got the card with $(a,b)$, produces new card with $(a/2,b/2)$, if both $a$ and $b$ are even and nothing in the opposite case; the third, having got the pair of cards with $(a,b)$ and $(b,c)$ produces new card with $(a,c)$. All the automates return the initial cards also. Suppose there was $(5,19)$ card initially. Is it possible to obtain a) $(1,50)$? b) $(1,100)$? c) Suppose there was $(a,b)$ card initially $(a<b)$. We want to obtain $(1,n)$ card. For what $n$ is it possible?

Two people $A$ and $B$ play the following game: They take from $\{0, 1, 2, 3,..., 1024\}$ alternately $512$, $256$, $128$, $64$, $32$, $16$, $8$, $4$, $2$, $1$, numbers away where $A$ first removes $512$ numbers, $B$ removes $256$ numbers etc. Two numbers $a, b$ remain ($a < b$). $B$ pays $A$ the amount $b - a$. $A$ would like to win as much as possible, $B$ would like to lose as little as possible. What profit does $A$ make if does every player play optimally according to their goals? The result must be justified.

[b]p1.[/b] While computing $7 - 2002 \cdot x$, John accidentally evaluates from left to right $((7 - 2002) \cdot x)$ instead of correctly using order of operations $(7 - (2002 \cdot x))$. If he gets the correct answer anyway, what is $x$? [b]p2.[/b] Given that $$x^2 + y^2 + z^2 = 6$$ $$ \left( \frac{x}{y} + \frac{y}{x} \right)^2 + \left( \frac{y}{z} + \frac{z}{y} \right)^2 + \left( \frac{z}{x} + \frac{x}{z} \right)^2 = 16.5,$$ what is $\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}$ ? [b]p3.[/b] Evaluate $$\frac{tan \frac{\pi}{4}}{4}+\frac{tan \frac{3\pi}{4}}{8}+\frac{tan \frac{5\pi}{4}}{16}+\frac{tan \frac{7\pi}{4}}{32}+ ...$$ [b]p4.[/b] Note that $2002 = 22 \cdot 91$, and so $2002$ is a multiple of the number obtained by removing its middle $2$ digits. Generalizing this, how many $4$-digit palindromes, $abba$, are divisible by the $2$-digit palindrome, $aa$? [b]p5.[/b] Let $ABCDE$ be a pyramid such that $BCDE$ is a square with side length $2$, and $A$ is $2$ units above the center of $BCDE$. If $F$ is the midpoint of $\overline{DE}$ and $G$ is the midpoint of $\overline{AC}$, what is the length of $\overline{DE}$? [b]p6.[/b] Suppose $a_1, a_2,..., a_{100}$ are real numbers with the property that $$i(a_1 + a_2 +... + a_i) = 1 + (a_{i+1} + a_{i+2} + ... + a_{100})$$ for all $i$. Compute $a_{10}$. [b]p7.[/b] A bug is sitting on one corner of a $3' \times 4' \times 5'$ block of wood. What is the minimum distance nit needs to travel along the block’s surface to reach the opposite corner? [b]p8.[/b] In the number game, a pair of positive integers $(n,m)$ is written on a blackboard. Two players then take turns doing the following: 1. If $n \ge m$, the player chooses a positive integer $c$ such that $n - cm \ge 0$, and replaces $(n,m)$ with $(n - cm,m)$. 2. If $m > n$, the player chooses a positive integer $c$ such that $m - cn \ge 0$, and replaces $(n,m)$ with $(n,m - cn)$. If $m$ or $n$ ever become $0$, the game ends, and the last player to have moved is declared the winner. If $(n,m)$ are originally $(20021000, 2002)$, what choices of $c$ are winning moves for the first player? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$n$ people are in the plane, so that the closest person is unique and each one shoot this closest person with a squirt gun. If $n$ is odd, prove that there exists at least one person that nobody shot. If $n$ is even, will there always be a person who escape? Justify that.

[b]p1.[/b] A tennis net is made of strings tied up together which make a grid consisting of small squares as shown below. [img]https://cdn.artofproblemsolving.com/attachments/9/4/72077777d57408d9fff0ea5e79be5ecb6fe8c3.png[/img] The size of the net is $100\times 10$ small squares. What is the maximal number of sides of small squares which can be cut without breaking the net into two separate pieces? (The side is cut only in the middle, not at the ends). [b]p2.[/b] What number is bigger $2^{300}$ or $3^{200}$ ? [b]p3.[/b] All noble knights participating in a medieval tournament in Camelot used nicknames. In the tournament each knight had combats with all other knights. In each combat one knight won and the second one lost. At the end of tournament the losers reported their real names to the winners and to the winners of their winners. Was there a person who knew the real names of all knights? [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $10$ rocks in the first pile and $12$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] There is an interesting $5$-digit integer. With a $1$ after it, it is three times as large as with a $1$ before it. What is the number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the occasion of the 47th Mathematical Olympiad 2016 the numbers 47 and 2016 are written on the blackboard. Alice and Bob play the following game: Alice begins and in turns they choose two numbers $a$ and $b$ with $a > b$ written on the blackboard, whose difference $a-b$ is not yet written on the blackboard and write this difference additionally on the board. The game ends when no further move is possible. The winner is the player who made the last move. Prove that Bob wins, no matter how they play. (Richard Henner)

[b]p1.[/b] What is the smallest positive integer $x$ such that $\frac{1}{x} <\sqrt{12011} - \sqrt{12006}$? [b]p2. [/b] Two soccer players run a drill on a $100$ foot by $300$ foot rectangular soccer eld. The two players start on two different corners of the rectangle separated by $100$ feet, then run parallel along the long edges of the eld, passing a soccer ball back and forth between them. Assume that the ball travels at a constant speed of $50$ feet per second, both players run at a constant speed of $30$ feet per second, and the players lead each other perfectly and pass the ball as soon as they receive it, how far has the ball travelled by the time it reaches the other end of the eld? [b]p3.[/b] A trapezoid $ABCD$ has $AB$ and $CD$ both perpendicular to $AD$ and $BC =AB + AD$. If $AB = 26$, what is $\frac{CD^2}{AD+CD}$ ? [b]p4.[/b] A hydrophobic, hungry, and lazy mouse is at $(0, 0)$, a piece of cheese at $(26, 26)$, and a circular lake of radius $5\sqrt2$ is centered at $(13, 13)$. What is the length of the shortest path that the mouse can take to reach the cheese that also does not also pass through the lake? [b]p5.[/b] Let $a, b$, and $c$ be real numbers such that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 3$. If $a^5 + b^5 + c^5\ne 0$, compute $\frac{(a^3+b^3+c^3)(a^4+b^4+c^4)}{a^5+b^5+c^5}$. [b]p6. [/b] Let $S$ be the number of points with integer coordinates that lie on the line segment with endpoints $\left( 2^{2^2}, 4^{4^4}\right)$ and $\left(4^{4^4}, 0\right)$. Compute $\log_2 (S - 1)$. [b]p7.[/b] For a positive integer $n$ let $f(n)$ be the sum of the digits of $n$. Calculate $$f(f(f(2^{2006})))$$ [b]p8.[/b] If $a_1, a_2, a_3, a_4$ are roots of $x^4 - 2006x^3 + 11x + 11 = 0$, find $|a^3_1 + a^3_2 + a^3_3 + a^3_4|$. [b]p9.[/b] A triangle $ABC$ has $M$ and $N$ on sides $BC$ and $AC$, respectively, such that $AM$ and $BN$ intersect at $P$ and the areas of triangles $ANP$, $APB$, and $PMB$ are $5$, $10$, and $8$ respectively. If $R$ and $S$ are the midpoints of $MC$ and $NC$, respectively, compute the area of triangle $CRS$. [b]p10.[/b] Jack's calculator has a strange button labelled ''PS.'' If Jack's calculator is displaying the positive integer $n$, pressing PS will cause the calculator to divide $n$ by the largest power of $2$ that evenly divides $n$, and then adding 1 to the result and displaying that number. If Jack randomly chooses an integer $k$ between $ 1$ and $1023$, inclusive, and enters it on his calculator, then presses the PS button twice, what is the probability that the number that is displayed is a power of $2$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In an $8$-square board -like the one in the figure- there is initially one checker in each square. $ \begin{tabular}{ | l | c | c |c | c| c | c | c | r| } \hline & & & & & & & \\ \hline \end{tabular} $ A move consists of choosing two tokens and moving one of them one square to the right and the other one one square to the left. If after $4$ moves the $8$ checkers are distributed in only $2$ boxes, determine what those boxes can be and how many checkers are in each one.

Ana and Bojan are playing a game: Ana chooses positive integers $a$ and $b$ and each one gets $2016$ pieces of paper, visible to both - Ana gets the pieces with the numbers $a+1$, $a+2$, $\ldots$, $a+2016$ and Bojan gets the pieces with the numbers $b+1$, $b+2$, $\ldots$, $b+2016$ on them. Afterwards, one of them writes the number $a+b$ on the board. In every move, Ana chooses one of her pieces of paper and hands it to Bojan who chooses one of his own, writes their sum on the board and removes them both from the game. When they run out of pieces, they multiply the numbers on the board together. If the result has the same remainder than $a+b$ when divided by $2017$, Bojan wins, otherwise, Ana wins. Who has the winning strategy?

Let $m$ and $n$ be positive integers. Player $A$ has a field of $m \times n$, and player $B$ has a $1 \times n$ field (the first is the number of rows). On the first move, each player places on each square of his field white or black chip as he pleases. At each next on the move, each player can change the color of randomly chosen pieces on your field to the opposite, provided that in no row for this move will not change more than one chip (it is allowed not to change not a single chip). The moves are made in turn, player $A$ starts. Player $A$ wins if there is such a position that in the only row player $B$'s squares, from left to right, are the same as in some row of player's field $A$. Prove that player $A$ has the ability to win for any game of player $B$ if and only if $n <2m$.