Given a $n\times n$ table, where $n$ is odd. There is either $1$ or $-1$ in its every field. A product of the numbers in the column is written under every column. A product of the numbers in the row is written to the right of every row. Prove that the sum of $2n$ products doesn't equal to $0$.

1) Cells of $8 \times 8$ table contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exists integer written in the same row or in the same column such that it is not relatively prime with $a$. Find maximum possible number of prime integers in the table. 2) Cells of $2n \times 2n$ table, $n \ge 2,$ contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exist integers written in the same row and in the same column such that they are not relatively prime with $a$. Find maximum possible number of prime integers in the table.

Given the polygons $P$ and $Q$ as shown in the grid below, cut $P$ into two polygons $P_1$ and $P_2$ such that, when pasted together differently, they form $Q$. [asy] import graph; size(16cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.05,xmax=15.10,ymin=-1.87,ymax=9.74; pen cqcqcq=rgb(0.75,0.75,0.75), zzttqq=rgb(0.6,0.2,0); draw((7,5)--(12,5)--(12,2)--(7,2)--cycle,zzttqq); draw((2,2)--(2,5)--(3,6)--(6,6)--(6,3)--(5,2)--cycle,zzttqq); /*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1; for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs); draw((0,8)--(0,0)); draw((0,0)--(13,0)); draw((13,0)--(13,8)); draw((13,8)--(0,8)); draw((7,5)--(12,5),zzttqq); draw((12,5)--(12,2),zzttqq); draw((12,2)--(7,2),zzttqq); draw((7,2)--(7,5),zzttqq); draw((2,2)--(2,5),zzttqq); draw((2,5)--(3,6),zzttqq); draw((3,6)--(6,6),zzttqq); draw((6,6)--(6,3),zzttqq); draw((6,3)--(5,2),zzttqq); draw((5,2)--(2,2),zzttqq); dot((0,0),linewidth(1pt)+ds); dot((13,0),linewidth(1pt)+ds); dot((0,8),linewidth(1pt)+ds); dot((2,2),linewidth(1pt)+ds); dot((6,6),linewidth(1pt)+ds); dot((13,8),linewidth(1pt)+ds); dot((7,2),linewidth(1pt)+ds); dot((7,5),linewidth(1pt)+ds); dot((12,2),linewidth(1pt)+ds); dot((12,5),linewidth(1pt)+ds); label("$Q$",(8.42,2.56),NE*lsf,zzttqq); dot((5,2),linewidth(1pt)+ds); dot((6,3),linewidth(1pt)+ds); dot((2,5),linewidth(1pt)+ds); dot((3,6),linewidth(1pt)+ds); label("$P$",(4.65,2.74),NE*lsf,zzttqq); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

In the Olympic school the exams are graded with whole numbers, the lowest possible grade is $0$, and the highest is $10$. In the arithmetic class the teacher takes two exams. This year he has $15$ students. When one of his students gets less than $3$ on the first exam and more than $7$ on the second exam, he calls him an overachieving student. The teacher, at the end of correcting the exams, averaged the $30$ grades and obtained $8$. What is the largest number of students who passed this class could have had?

Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$, and for $n \ge 7$: $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$ where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$. For example : ${\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$. For all integers $n \ge 2$, let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$. The minus $-$ denotes the ''[i]remove it if it is there[/i]'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$. However $S_5= \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$. 1. Determine $S_7,S_8,S_9$ and $S_{10}$. 2. We say that a set $S_n$ for $n\ge 6$ is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$. Prove that $S_7,S_8,S_9$ and $S_{10}$ are well-balanced . 3. Is the set $S_{2019}$ well-balanced? Justify your answer.

Let $M$ be a finite set and $P=\{ M_1,M_2,\ldots ,M_l\}$ a partition of $M$ (i.e., $\bigcup_{i=1}^k M_i, M_i\not=\emptyset, M_i\cap M_j =\emptyset$ for all $i,j\in\{1,2, \ldots ,k\} ,i\not= j)$. We define the following elementary operation on $P$: Choose $i,j\in\{1,2,\ldots ,k\}$, such that $i=j$ and $M_i$ has a elements and $M_j$ has $b$ elements such that $a\ge b$. Then take $b$ elements from $M_i$ and place them into $M_j$, i.e., $M_j$ becomes the union of itself and a $b$-element subset of $M_i$, while the same subset is subtracted from $M_i$ (if $a=b$, $M_i$ is thus removed from the partition). Let a finite set $M$ be given. Prove that the property “for every partition $P$ of $M$ there exists a sequence $P=P_1,P_2,\ldots ,P_r$ such that $P_{i+1}$ is obtained from $P_i$ by an elementary operation and $P_r=\{M\}$” is equivalent to “the number of elements of $M$ is a power of $2$.”

Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a fleet to which no new ship can be added.

It is known that there are $3$ buildings in the same shape which are located in an equilateral triangle. Each building has a $2015$ floor with each floor having one window. In all three buildings, every $1$st floor is uninhabited, while each floor of others have exactly one occupant. All windows will be colored with one of red, green or blue. The residents of each floor of a building can see the color of the window in the other buildings of the the same floor and one floor just below it, but they cannot see the colors of the other windows of the two buildings. Besides that, sresidents cannot see the color of the window from any floor in the building itself. For example, resident of the $10$th floor can see the colors of the $9$th and $10$th floor windows for the other buildings (a total of $4$ windows) and he can't see the color of the other window. We want to color the windows so that each resident can see at lest $1$ window of each color. How many ways are there to color those windows?

Let $m,n,k$ be pairwise relatively prime positive integers greater than $3$. Find the minimal possible number of points on the plane with the following property: there are $x$ of them which are the vertices of a regular $x$-gon for $x = m, x = n, x = k$. (E.Piryutko)

A $20 \times 20$ treasure map is glued to a torus. A treasure is hidden in a cell of this map. We can ask questions about $1\times 4$ or $4 \times 1$ rectangles so that we find out if there is a treasure in this rectangle or not. The answers to all questions are absolutely true, but they are given only after all rectangles we want to ask are set. What is the least amount of questions needed to be asked so that we can be sure to find the treasure? (If you describe the position of the cells in a torus with numbers $(i, j)$ of row and column, $1 \leq i, j \leq 20$, then two cells are neighbors, if and only if two of the coordinates they have are the same, and the other two differ by $1$ mod $20$.)

Two kids $A$ and $B$ play a game as follows: from a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n - 1$, inclusive. 3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive. The winner is the one who takes the last marble. Determine all natural numbers $n$ for which $A$ has a winning strategy

You have a $10\times 10$ grid of squares. You write a number in each square as follows: you write $1$, $2$, $3$ ,$ ...$ ,$10$ from left to right across the top row, then $11$, $12$, $...$, $20$ across the second row, and so on, ending with a $100$ in the bottom right square. You then write a second number in each square, writing $1$, $2$, $3$ ,$ ...$ ,$10$ in the first column (from top to bottom), then $11$, $12$, $...$, $20$ in the second column, and so forth. When this process is finished, how many squares will have the property that their two numbers sum to $101$?

Fix a set of integers $S$. An integer is [i]clean[/i] if it is the sum of distinct elements of $S$ in exactly one way, and [i]dirty[/i] otherwise. Prove that the set of dirty numbers is either empty or infinite. [i]Note:[/i] We consider the empty sum to equal \(0\). [i]Proposed by Tony Wang and Ethan Tan[/i]

$ S$ is the set of all sequences $ \{a_i| 1 \leq i \leq 7, a_i \equal{} 0 \text{ or } 1\}.$ The distance between two elements $ \{a_i\}$ and $ \{b_i\}$ of $ S$ is defined as \[ \sum^7_{i \equal{} 1} |a_i \minus{} b_i|. \] $ T$ is a subset of $ S$ in which any two elements have a distance apart greater than or equal to 3. Prove that $ T$ contains at most 16 elements. Give an example of such a subset with 16 elements.

Each two of $n$ students, who attended an activity, have different ages. It is given that each student shook hands with at least one student, who did not shake hands with anyone younger than the other. Find all possible values of $n$.

p1. Circle $M$ is the incircle of ABC, while circle $N$ is the incircle of $ACD$. Circles $M$ and $N$ are tangent at point $E$. If side length $AD = x$ cm, $AB = y$ cm, $BC = z$ cm, find the length of side $DC$ (in terms of $x, y$, and $z$). [img]https://cdn.artofproblemsolving.com/attachments/d/5/66ddc8a27e20e5a3b27ab24ff1eba3abee49a6.png[/img] p2. The address of the house on Jalan Bahagia will be numbered with the following rules: $\bullet$ One side of the road is numbered with consecutive even numbers starting from number $2$. $\bullet$ The opposite side is numbered with an odd number starting from number $3$. $\bullet$ In a row of even numbered houses, there is some land vacant house that has not been built. $\bullet$ The first house numbered $2$ has a neighbor next door. When the RT management ordered the numbers of the house, it is known that the cost of making each digit is $12.000$ Rp. For that, the total cost to be incurred is $1.020.000$ Rp. It is also known that the cost of all even-sided house numbers is $132.000$ Rp. cheaper than the odd side. When the land is empty later a house has been built, the number of houses on the even and odd sides is the same. Determine the number of houses that are now on Jalan Bahagia . p3. Given the following problem: Each element in the set $A = \{10, 11, 12,...,2008\}$ multiplied by each element in the set $B = \{21, 22, 23,...,99\}$. The results are then added together to give value of $X$. Determine the value of $X$. Someone answers the question by multiplying $2016991$ with $4740$. How can you explain that how does that person make sense? p4. Let $P$ be the set of all positive integers between $0$ and $2008$ which can be expressed as the sum of two or more consecutive positive integers . (For example: $11 = 5 + 6$, $90 = 29 + 30 + 31$, $100 = 18 + 19 +20 + 21 + 22$. So $11, 90, 100$ are some members of $P$.) Find the sum of of all members of $P$. p5. A four-digit number will be formed from the numbers at $0, 1, 2, 3, 4, 5$ provided that the numbers in the number are not repeated, and the number formed is a multiple of $3$. What is the probability that the number formed has a value less than $3000$?

Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that: [i](i)[/i] $i$ and $n - i$ always receive the same color, and [i](ii)[/i] for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$ Prove that all numbers in $N$ must receive the same color.

A $ 9 \times 9$ square is divided into unit squares. Is it possible to fill each unit square with a number $ 1, 2,..., 9$ in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?

Suppose that $ p_1,p_2,p_3,q_1,q_2,q_3$ are six points in the plane and that the distance between $ p_i$ and $ q_j$ ($ i,j \equal{} 1,2,3$) is $ i \plus{} j$. Show that the six points are collinear.

Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves.

Find a way of arranging $n$ girls and $n$ boys around a round table for which $d_n-c_n$ is maximum, where dn is the number of girls sitting between two boys and $c_n$ is the number of boys sitting between two girls.

For an integer $n>2$, the tuple $(1, 2, \ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\{1, 2,\ldots, n\}$ can appear on the blackboard in this way.

Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number $1$, an arc between a red and a blue point is assigned a number $2$, and an arc between a blue and a green point is assigned a number $3$. The arcs between two points of the same colour are assigned a number $0$. What is the greatest possible sum of all the numbers assigned to the arcs?

If $a$ and $b$ are selected uniformly from $\{0,1,\ldots,511\}$ without replacement, the expected number of $1$'s in the binary representation of $a+b$ can be written in simplest from as $\tfrac{m}{n}$. Compute $m+n$.