$99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today? [i]Proposed by N. Vlasova, S. Berlov[/i]

A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other. Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges $\overrightarrow{uv}$ and $\overrightarrow{vw}$, those two edges are in different colors. Note that it is permissible for $\overrightarrow{uv}$ and $\overrightarrow{uw}$ to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each $n$, determine the minimum directed-edge-chromatic-number over all tournaments on $n$ vertices. [i]Proposed by Po-Shen Loh[/i]

A star $K_{1,3}$ is called a paw. suppose that $G$ is a graph without any induced paws. prove that $\chi(G)\le(\omega(G))^2$. (15 points)

In country some cities are connected by oneway flights( There are no more then one flight between two cities). City $A$ called "available" for city $B$, if there is flight from $B$ to $A$, maybe with some transfers. It is known, that for every 2 cities $P$ and $Q$ exist city $R$, such that $P$ and $Q$ are available from $R$. Prove, that exist city $A$, such that every city is available for $A$.

[b]p1.[/b] Anne purchased yesterday at WalMart in Puerto Rico $6$ identical notebooks, $8$ identical pens and $7$ identical erasers. Anne remembers that each eraser costs $73$ cents. She did not buy anything else. Anne told her mother that she spent $12$ dollars and $76$ cents at Walmart. Can she be right? Note that in Puerto Rico there is no sales tax. [b]p2.[/b] Two men ski one after the other first in a flat field and then uphill. In the field the men run with the same velocity $12$ kilometers/hour. Uphill their velocity drops to $8$ kilometers/hour. When both skiers enter the uphill trail segment the distance between them is $300$ meters less than the initial distance in the field. What was the initial distance between skiers? (There are $1000$ meters in 1 kilometer.) [b]p3.[/b] In the equality $** + **** = ****$ all the digits are replaced by $*$. Restore the equality if it is known that any numbers in the equality does not change if we write all its digits in the opposite order. [b]p4.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started: ”-None of you has $8$ legs. Only I have 8 legs!” Which polyleg has exactly $8$ legs? [b]p5.[/b] Cut the figure shown below in two equal pieces. (Both the area and the form of the pieces must be the same.) [img]https://cdn.artofproblemsolving.com/attachments/e/4/778678c1e8748e213ffc94ba71b1f3cc26c028.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. Suppose we are given $2^n+1$ distinct sets, each containing finitely many objects. Place each set into one of two categories, the red sets and the blue sets, so that there is at least one set in each category. We define the [i]symmetric difference[/i] of two sets as the set of objects belonging to exactly one of the two sets. Prove that there are at least $2^n$ different sets which can be obtained as the symmetric difference of a red set and a blue set.

Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition: For any three elements in $M$, there exist two of them $a$ and $b$ such that $a|b$ or $b|a$. Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$

Let $n$ a positive integer. In a $2n\times 2n$ board, $1\times n$ and $n\times 1$ pieces are arranged without overlap. Call an arrangement [b]maximal[/b] if it is impossible to put a new piece in the board without overlapping the previous ones. Find the least $k$ such that there is a [b]maximal[/b] arrangement that uses $k$ pieces.

Given $n$ chairs around a circle which are marked with numbers from 1 to $n$ .There are $k$, $k \leq 4 \cdot n$ students sitting on those chairs .Two students are called neighbours if there is no student sitting between them. Between two neighbours students ,there are at less 3 chairs. Find the number of choices of $k$ chairs so that $k$ students can sit on those and the condition is satisfied.

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

From each of the vertices of a regular $n$-gon a car starts to move with constant speed along the perimeter of the $n$-gon in the same direction. Prove that if all the cars end up at a vertex $A$ at the same time, then they never again meet at any other vertex of the $n$-gon. Can they meet again at $A \ ?$

Find the maximum value of an integer $B$ such that for every 9 distinct natural number with the sum of $2023$, there must exist a sum of 4 of the number that is greater than or equal to $B$

Coins are placed in some squares on a $n\times n$ board. Each coin can be moved towards the square symmetrical with respect to either of the two diagonals, as long as that square is empty. The initial coin setup is said to be [i]good [/i], if any coin can make the first move. (a) Determine the maximum number of coins $M$ that can be placed on the $n\times n$ board, such that the configuration is good. (b) Calculate the total number of good configurations that have exactly $M$ coins.

On a round table, $n$ bowls are arranged in a circle. Anja walks around the table clockwise, placing marbles in the bowls according to the following rule: She places a marble in any first bowl, then goes one bowl further and puts a marble in there. Then she goes two shells before putting another marble, then she goes three shells, etc. If there is at least one marble in each shell, she stops. For which $n$ does this occur?

There are $2007$ boys and $2007$ girls in a middle school,every student can attend no more than $100$ academic meetings,if we know any pair of students with different sex attend at least one common meeting.prove that there must be a meeting with at least $11$ boys and $11$ girls attend.

We are given a $4\times 4$ table. a) Place $7$ stars in the cells in such a way that the erasing of any two rows and two columns will leave at least one of the stars. b) Prove that if there are less than $7$ stars, you can always find two columns and two rows such that erasing them, no star remains in the table.

The numbers $1,\, 2,\, \ldots\, , n^{2}$ are written in the squares of an $n \times n$ board in some order. Initially there is a token on the square labelled with $n^{2}.$ In each step, the token can be moved to any adjacent square (by side). At the beginning, the token is moved to the square labelled with the number $1$ along a path with the minimum number of steps. Then it is moved to the square labelled with $2,$ then to square $3,$ etc, always taking the shortest path, until it returns to the initial square. If the total trip takes $N$ steps, find the smallest and greatest possible values of $N.$

Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

Antonio and Beltran have impeccable logical reasoning, they put on a hat with a integer between $0$ and $19$ (including both) so that each of them sees the number that has the other (but cannot see his own number), and they must try to guess the number that have on their hat. They have a timer that a bell rings every minute and the moment it rings. This is when they must say if they know the number on their hat. A third person tells them: ''the sum of the numbers is $6$ or $11$ or $19$''. At that moment it begins to run time. After a minute the bell rings and neither of them says anything. The second minute passes , the doorbell rings and neither of us says anything. Time continues to pass and when the bell rings for the tenth time Antonio says that he already knows what is his number. Just determine the number each has in his hat.

Let $n$ blue lines, no two of which are parallel and no three concurrent, be drawn on a plane. An intersection of two blue lines is called a blue point. Through any two blue points that have not already been joined by a blue line, a red line is drawn. An intersection of two red lines is called a red point, and an intersection of red line and a blue line is called a purple point. What is the maximum possible number of purple points?

[b]p1.[/b] The number $100$ is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum. [b]p2.[/b] Inside an equilateral triangle of side length $s$ are three mutually tangent circles of radius $1$, each one of which is also tangent to two sides of the triangle, as depicted below. Find $s$. [img]https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png[/img] [b]p3.[/b] Color a $4\times 7$ rectangle so that each of its $28$ unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color. [b]p4.[/b] (a) Show that the $y$-intercept of the line through any two distinct points of the graph of $f(x) = x^2$ is $-1$ times the product of the $x$-coordinates of the two points. (b) Find all real valued functions with the property that the $y$-intercept of the line through any two distinct points of its graph is $-1$ times the product of the $x$-coordinates. Prove that you have found all such functions and that all functions you have found have this property. [b]p5.[/b] Let $n$ be a positive integer. We consider sets $A \subseteq \{1, 2,..., n\}$ with the property that the equation $x+y=z$ has no solution with $x\in A$, $y \in A$, $z \in A$. (a) Show that there is a set $A$ as described above that contains $[(n + l)/2]$ members where $[x]$ denotes the largest integer less than or equal to $x$. (b) Show that if $A$ has the property described above, then the number of members of $A$ is less than or equal to $[(n + l)/2]$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square. [b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$. [b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img] [b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ 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]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$. Find the maximum value of $n$.

In a chess tournament there are $n \geq 5$ players, and they have already played $\left[ \frac{n^2}{4} \right] +2$ games (each pair have played each other at most once). [b](a)[/b] Prove that there are five players $a, b, c, d, e$ for which the pairs $ab, ac, bc, ad, ae, de$ have already played. [b](b)[/b] Is the statement also valid for the $\left[ \frac{n^2}{4} \right] +1$ games played? Make the proof by induction over $n.$

$6$ open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all $6$ disks.