Two players in turns color the sides of an $n$-gon. The first player colors any side that has $0$ or $2$ common vertices with already colored sides. The second player colors any side that has exactly $1$ common vertex with already colored sides. The player who cannot move, loses. For which $n$ the second player has a winning strategy?

A network is a simple directed graph such that each edge $ e$ has two intger lower and upper capacities $ 0\leq c_l(e)\leq c_u(e)$. A circular flow on this graph is a function such that: 1) For each edge $ e$, $ c_l(e)\leq f(e)\leq c_u(e)$. 2) For each vertex $ v$: \[ \sum_{e\in v^\plus{}}f(e)\equal{}\sum_{e\in v^\minus{}}f(e)\] a) Prove that this graph has a circular flow, if and only if for each partition $ X,Y$ of vertices of the network we have: \[ \sum_{\begin{array}{c}{e\equal{}xy}\\{x\in X,y\in Y}\end{array}} c_l(e)\leq \sum_{\begin{array}{c}{e\equal{}yx}\\{y\in Y,x\in X}\end{array}} c_l(e)\] b) Suppose that $ f$ is a circular flow in this network. Prove that there exists a circular flow $ g$ in this network such that $ g(e)\equal{}\lfloor f(e)\rfloor$ or $ g(e)\equal{}\lceil f(e)\rceil$ for each edge $ e$.

Nineteen weights of mass $1$ gm, $2$ gm, $3$ gm, . . . , $19$ gm are given. Nine are made of iron, nine are of bronze and one is pure gold. It is known that the total mass of all the iron weights is $90$ gm more than the total mass of all the bronze ones. Find the mass of the gold weight . (V Proizvolov)

$a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ are $2n$ positive real numbers such that $a_1,a_2,\cdots ,a_n$ aren't all equal. And assume that we can divide $a_1,a_2,\cdots ,a_n$ into two subsets with equal sums.similarly $b_1,b_2,\cdots ,b_n$ have these two conditions. Prove that there exist a simple $2n$-gon with sides $a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ and parallel to coordinate axises Such that the lengths of horizontal sides are among $a_1,a_2,\cdots ,a_n$ and the lengths of vertical sides are among $b_1,b_2,\cdots ,b_n$.(simple polygon is a polygon such that it doesn't intersect itself)

$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.

[u]Round 1 [/u] [b]p1. [/b]Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the 12th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p2.[/b] Show that in the sequence $10017$, $100117$, $1001117$, $...$ all numbers are divisible by $53$. [b]p3.[/b] Harry and Draco have three wands: a bamboo wand, a willow wand, and a cherry wand, all of the same length. They must perform a spell wherein they take turns picking a wand and breaking it into three parts – first Harry, then Draco, then Harry again. But in order for the spell to work, Harry has to make sure it is possible to form three triangles out of the pieces of the wands, where each triangle has a piece from each wand. How should he break the wands to ensure the success of the spell? [b]p4.[/b] A $2\times 2\times 2$ cube has $4$ equal squares on each face. The squares that share a side are called neighbors (thus, each square has $4$ neighbors – see picture). Is it possible to write an integer in each square in such a way that the sum of each number with its $4$ neighbors is equal to $13$? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/8/4/0f7457f40be40398dee806d125ba26780f9d3a.png[/img] [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2 [/u] [b]p6.[/b] A red square is placed on a table. $2010$ white squares, each the same size as the red square, are then placed on the table in such a way that the red square is fully covered and the sides of every white square are parallel to the sides of the red square. Is it always possible to remove one of the white squares so the red square remains completely covered? [b]p7.[/b] A computer starts with a given positive integer to which it randomly adds either $54$ or $77$ every second and prints the resulting sum after each addition. For example, if the computer is given the number $1$, then a possible output could be: $1$, $55$, $109$, $186$, $…$ Show that after finitely many seconds the computer will print a number whose last two digits are the same. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Ephram Chun is a senior and math captain at Lexington High School. He is well-loved by the freshmen, who seem to only listen to him. Other than being the father figure that the freshmen never had, Ephramis also part of the Science Bowl and Science Olympiad teams along with being part of the highest orchestra LHS has to offer. His many hobbies include playing soccer, volleyball, and the many forms of chess. We hope that he likes the questions that we’ve dedicated to him! [b]p1.[/b] Ephram is scared of freshmen boys. How many ways can Ephram and $4$ distinguishable freshmen boys sit together in a row of $5$ chairs if Ephram does not want to sit between $2$ freshmen boys? [b]p2.[/b] Ephram, who is a chess enthusiast, is trading chess pieces on the black market. Pawns are worth $\$100$, knights are worth $\$515$, and bishops are worth $\$396$. Thirty-four minutes ago, Ephrammade a fair trade: $5$ knights, $3$ bishops, and $9$ rooks for $8$ pawns, $2$ rooks, and $11$ bishops. Find the value of a rook, in dollars. [b]p3.[/b] Ephramis kicking a volleyball. The height of Ephram’s kick, in feet, is determined by $$h(t) = - \frac{p}{12}t^2 +\frac{p}{3}t ,$$ where $p$ is his kicking power and $t$ is the time in seconds. In order to reach the height of $8$ feet between $1$ and $2$ seconds, Ephram’s kicking power must be between reals $a$ and $b$. Find is $100a +b$. [b]p4.[/b] Disclaimer: No freshmen were harmed in the writing of this problem. Ephram has superhuman hearing: He can hear sounds up to $8$ miles away. Ephramstands in the middle of a $8$ mile by $24$ mile rectangular grass field. A freshman falls from the sky above a point chosen uniformly and randomly on the grass field. The probability Ephram hears the freshman bounce off the ground is $P\%$. Find $P$ rounded to the nearest integer. [img]https://cdn.artofproblemsolving.com/attachments/4/4/29f7a5a709523cd563f48176483536a2ae6562.png[/img] [b]p5.[/b] Ephram and Brandon are playing a version of chess, sitting on opposite sides of a $6\times 6$ board. Ephram has $6$ white pawns on the row closest to himself, and Brandon has $6$ black pawns on the row closest to himself. During each player’s turn, their only legal move is to move one pawn one square forward towards the opposing player. Pawns cannot move onto a space occupied by another pawn. Players alternate turns, and Ephram goes first (of course). Players take turns until there are no more legal moves for the active player, at which point the game ends. Find the number of possible positions the game can end in. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Juku has the first $100$ volumes of the Harrie Totter book series at his home. For every$ i$ and $j$, where $1 \le i < j \le 100$, call the pair $(i, j)$ reversed if volume No. $j$ is before volume No, $i$ on Juku’s shelf. Juku wants to arrange all volumes of the series to one row on his shelf in such a way that there does not exist numbers $i, j, k$, where $1 \le i < j < k \le 100$, such that pairs $(i, j)$ and $(j, k)$ are both reversed. Find the largest number of reversed pairs that can occur under this condition

What is the maximum number of checkers it is possible to put on a $ n \times n$ chessboard such that in every row and in every column there is an even number of checkers?

At the faculty of mathematics study $40$ boys and $10$ girls. Every girl acquaintance with all boys, who older than her, or tall(higher) than her. Prove that there exist two boys such that the sets of acquainted-girls of the boys are same.

[b]p1.[/b] Fill in each box with an integer from $1$ to $9$. Each number in the right column is the product of the numbers in its row, and each number in the bottom row is the product of the numbers in its column. Some numbers may be used more than once, and not every number from $1$ to $9$ is required to be used. [img]https://cdn.artofproblemsolving.com/attachments/c/0/0212181d87f89aac374f75f1f0bde6d0600037.png[/img] [b]p2.[/b] A set $X$ is called [b]prime-difference free [/b] (henceforth pdf) if for all $x, y \in X$, $|x - y|$ is not prime. Find the number n such that the following both hold. $\bullet$ There is a pdf set of size $n$ that is a subset of $\{1,..., 2016\}$, and $\bullet$ There is no pdf set of size $n + 1$ that is a subset of $\{1,..., 2016\}$. [b]p3.[/b] Let $X_1,...,X_{15}$ be a sequence of points in the $xy$-plane such that $X_1 = (10, 0)$ and $X_{15} = (0, 10)$. Prove that for some $i \in \{1, 2,..., 14\}$, the midpoint of $X_iX_{i+1}$ is of distance greater than $1/2$ from the origin. [b]p4.[/b] Suppose that $s_1, s_2,..., s_{84}$ is a sequence of letters from the set $\{A,B,C\}$ such that every four-letter sequence from $\{A,B,C\}$ occurs exactly once as a consecutive subsequence $s_k$, $s_{k+1}$, $s_{k+2}$, $s_{k+3}$. Suppose that $$(s_1, s_2, s_3, s_4, s_5) = (A,B,B,C,A).$$ What is $s_{84}$? Prove your answer. [b]p5.[/b] Determine (with proof) whether or not there exists a sequence of positive real numbers $a_1, a_2, a_3,...$ with both of the following properties: $\bullet$ $\sum^n_{i=1} a_i \le n^2$, for all $n \ge 1$, and $\bullet$ $\sum^n_{i=1} \frac{1}{a_i} \le 2016$, for all $n \ge 1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive real numbers are arranged in the form: $ 1 \ \ \ 3 \ \ \ 6 \ \ \ 10 \ \ \ 15 ...$ $ 2 \ \ \ 5 \ \ \ 9 \ \ \ 14 ...$ $ 4 \ \ \ 8 \ \ \ 13 ...$ $ 7 \ \ \ 12 ...$ $ 11 ...$ Find the number of the line and column where the number 2002 stays.

The fence consists of $25$ vertical bars. The heights of the bars are pairwise distinct positive integers from $1$ to $25$. The width of every bar is $1$. Find the maximum $S$ for which regardless of the order of the bars one can find a rectangle of area $S$ formed by the fence.

A competition room of HOMC has $m \times n$ students where $m, n$ are integers larger than $2$. Their seats are arranged in $m$ rows and $n$ columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally $27$ handshakes. Find the number of students in the room.

In a permutation $(x_1, x_2, \dots , x_n)$ of the set $1, 2, \dots , n$ we call a pair $(x_i, x_j )$ discordant if $i < j$ and $x_i > x_j$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3).$

A permutation $ \{x_1, x_2, \ldots, x_{2n}\}$ of the set $ \{1,2, \ldots, 2n\}$ where $ n$ is a positive integer, is said to have property $ T$ if $ |x_i \minus{} x_{i \plus{} 1}| \equal{} n$ for at least one $ i$ in $ \{1,2, \ldots, 2n \minus{} 1\}.$ Show that, for each $ n$, there are more permutations with property $ T$ than without.

Given an integer $n\geq 2$. There are $N$ distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersection. Initially there is a coin on each of the intersection points of the circles. Starting from $X$, players $X$ and $Y$ alternatively take away a coin, with the restriction that one cannot take away a coin lying on the same circle as the last coin just taken away by the opponent in the previous step. The one who cannot do so will lost. In particular, one loses where there is no coin left. For what values of $n$ does $Y$ have a winning strategy?

In a regular $ 6n\plus{}1$-gon, $ k$ vertices are painted in red and the others in blue. Prove that the number of isosceles triangles whose vertices are of the same color does not depend on the arrangement of the red vertices.

Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.

There are $n$ pawns on $n$ distinct squares of a $19\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely?

Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$. A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$. A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$. Now put a pawn on $(0, 0)$. You can make a (nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B. Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.

Consider a $6 \times 6$ grid. Define a [i]diagonal[/i] to be the six squares whose coordinates $(i,j)$ ($1 \le i,j \le 6)$ satisfy $i-j \equiv k \pmod 6$ for some $k=0,1,\dots,5$. Hence there are six diagonals. Determine if it is possible to fill it with the numbers $1,2,\dots,36$ (each exactly once) such that each row, each column, and each of the six diagonals has the same sum.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $ n$ be a positive integer. A $4\times n$ rectangular grid is divided in$ 2\times 1$ or $1\times 2$ rectangles (as if it were completely covered with tiles of domino, no overlaps or gaps). Then all the grid points which are vertices of one of the $2\times 1$ or $1\times 2$ rectangles, are painted red. What is the least amount of red points you can get?

A $4$ x $4$ square board is called $brasuca$ if it follows all the conditions: • each box contains one of the numbers $0, 1, 2, 3, 4$ or $5$; • the sum of the numbers in each line is $5$; • the sum of the numbers in each column is $5$; • the sum of the numbers on each diagonal of four squares is $5$; • the number written in the upper left box of the board is less than or equal to the other numbers the board; • when dividing the board into four $2$ × $2$ squares, in each of them the sum of the four numbers is $5$. How many $"brasucas"$ boards are there?