A regular polygon with $1000$ sides has the vertices colored in red, yellow or blue. A move consists in choosing to adjiacent vertices colored differently and coloring them in the third color. Prove that there is a sequence of moves after which all the vertices of the polygon will have the same color. Marius Ghergu

Given a positive integer $n$, Susann fills a square of $n \times n$ boxes. In each box she inscribes an integer, taking care that each row and each column contains distinct numbers. After this an imp appears and destroys some of the boxes. Show that Susann can choose some of the remaining boxes and colour them red, satisfying the following two conditions: 1) There are no two red boxes in the same column or in the same row. 2) For each box which is neither destroyed nor coloured, there is a red box with a larger number in the same row or a red box with a smaller number in the same column. [i]Proposed by Christian Reiher[/i]

There are $11$ committees in a club. Each committee has $5$ members and every two committees have a member in common. Show that there is a member who belongs to $4$ committees.

There are two letter sequences $A$ and $B$, both with length $100$ letters. In one move you can insert in any place of sequence ( possibly to start or to end) any number of same letters or remove any number of consecutive same letters. Prove that it is possible to make second sequence from first sequence using not more than $100$ moves.

Let $S$ be the set of all non-increasing sequences of numbers $a_1 \geq a_2 \geq \ldots \geq a_{101}$ such that $a_i \in \{ 0,1,\ldots ,101 \}$ for all $1 \leq i \leq 101$ For every sequence $s \in S$ let $$f(s)=\lceil \frac{a_1}{2} \rceil+\lfloor \frac{a_2}{2} \rfloor + \lceil \frac{a_3}{2} \rceil + \ldots + \lfloor \frac{a_{100}}{2} \rfloor + \lceil \frac{a_{101}}{2} \rceil$$ where $\lfloor x \rfloor$ is the greatest integer, not exceeding $x$, and $\lceil x \rceil$ is the least integer at least $x$. Prove that the number of sequences $s \in S$ for which $f(s)$ is even is the same, as the number of sequences $s$ for which $f(s)$ is odd [i]M. Zorka[/i]

Given are two polyominos, the first one is an L-shape consisting of three squares, the other one contains at least two squares. Prove that if $n$ and $m$ are coprime then at most one of the $n\times n$ and $m\times m$ boards can be tiled by translated copies of the two polyominos. [i]Proposed by: András Imolay, Dávid Matolcsi, Ádám Schweitzer and Kristóf Szabó, Budapest[/i]

The numbers $1,\frac12,\frac13,\ldots,\frac1{1999}$ are written on a blackboard. In every step, we choose two of them, say $a$ and $b$, erase them, and write the number $ab+a+b$ instead. This step is repeated until only one number remains. Can the last remaining number be equal to $2000$?

Let $n > 1$ be an integer. A set $S \subseteq \{ 0, 1, 2, \cdots , 4n - 1 \}$ is called ’sparse’ if for any $k \in \{ 0, 1, 2, \cdots , n - 1 \}$ the following two conditions are satisfied: $(a)$ The set $S \cap \{4k - 2, 4k - 1, 4k, 4k + 1, 4k + 2 \}$ has at most two elements; $(b)$ The set $S \cap \{ 4k +1, 4k +2, 4k +3 \}$ has at most one element. Prove that there are exactly $8 \cdot 7^{n-1}$ sparse subsets.

It is given $5$ numbers $1$, $3$, $5$, $7$, $9$. We get the new $5$ numbers such that we take arbitrary $4$ numbers(out of current $5$ numbers) $a$, $b$, $c$ and $d$ and replace them with $\frac{a+b+c-d}{2}$, $\frac{a+b-c+d}{2}$, $\frac{a-b+c+d}{2}$ and $\frac{-a+b+c+d}{2}$. Can we, with repeated iterations, get numbers: $a)$ $0$, $2$, $4$, $6$ and $8$ $b)$ $3$, $4$, $5$, $6$ and $7$

$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.) Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?

There are $100$ red, $100$ yellow and $100$ green sticks. One can construct a triangle using any three sticks all of different colours (one red, one yellow and one green). Prove that there is a colour such that one can construct a triangle using any three sticks of this colour.

A sequence $a_1, a_2, a_3, . . . , a_{100}$ of $100$ (not necessarily distinct) positive numbers satisfy that the$ 99$ fractions$$\frac{a_1}{a_2},\frac{a_2}{a_3},\frac{a3}{a_4}, ... ,\frac{a_{99}}{a_{100}}$$ are all distinct. How many distinct numbers must there be, at least, in the sequence $a_1, a_2, a_3, . . . , a_{100}$?

A set of $n$ distinct integer weights $w_1,w_2,\ldots, w_n$ is said to be [i]balanced[/i] if after removing any one of weights, the remaining $(n-1)$ weights can be split into two subcollections (not necessarily with equal size)with equal sum. $a)$ Prove that if there exist [i]balanced[/i] sets of sizes $k,j$ then also a [i]balanced[/i] set of size $k+j-1$. $b)$ Prove that for all [i]odd[/i] $n\geq 7$ there exist a [i]balanced[/i] set of size $n$.

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

[b]p1.[/b] An organization decides to raise funds by holding a $\$60$ a plate dinner. They get prices from two caterers. The first caterer charges $\$50$ a plate. The second caterer charges according to the following schedule: $\$500$ set-up fee plus $\$40$ a plate for up to and including $61$ plates, and $\$2500$ $\log_{10}\left(\frac{p}{4}\right)$ for $p > 61$ plates. a) For what number of plates $N$ does it become at least as cheap to use the second caterer as the first? b) Let $N$ be the number you found in a). For what number of plates $X$ is the second caterer's price exactly double the price for $N$ plates? c) Let $X$ be the number you found in b). When X people appear for the dinner, how much profit does the organization raise for itself by using the second caterer? [b]p2.[/b] Let $N$ be a positive integer. Prove the following: a) If $N$ is divisible by $4$, then $N$ can be expressed as the sum of two or more consecutive odd integers. b) If $N$ is a prime number, then $N$ cannot be expressed as the sum of two or more consecutive odd integers. c) If $N$ is twice some odd integer, then $N$ cannot be expressed as the sum of two or more consecutive odd integers. [b]p3.[/b] Let $S =\frac{1}{1^2} +\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...$ a) Find, in terms of $S$, the value of $S =\frac{1}{2^2} +\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...$ b) Find, in terms of $S$, the value of$S =\frac{1}{1^2} +\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+...$ c) Find, in terms of $S$, the value of$S =\frac{1}{1^2} -\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+...$ [b]p4.[/b] Let $\{P_1, P_2, P_3, ...\}$ be an infinite set of points on the $x$-axis having positive integer coordinates, and let $Q$ be an arbitrary point in the plane not on the $x$-axis. Prove that infinitely many of the distances $|P_iQ|$ are not integers. a) Draw a relevant picture. b) Provide a proof. [b]p5.[/b] Point $P$ is an arbitrary point inside triangle $ABC$. Points $X$, $Y$ , and $Z$ are constructed to make segments $PX$, $PY$ , and $PZ$ perpendicular to $AB$, $BC$, and $CA$, respectively. Let $x$, $y$, and $z$ denote the lengths of the segments $PX$, $PY$ , and $PZ$, respectively. a) If triangle $ABC$ is an equilateral triangle, prove that $x + y + z$ does not change regardless of the location of $P$ inside triangle ABC. b) If triangle $ABC$ is an isosceles triangle with $|BC| = |CA|$, prove that $x + y + z$ does not change when $P$ moves along a line parallel to $AB$. c) Now suppose that triangle $ABC$ is scalene (i.e., $|AB|$, $|BC|$, and $|CA|$ are all different). Prove that there exists a line for which $x+y+z$ does not change when $P$ moves along this line. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We color some unit squares in a $ 99\times 99 $ square grid with one of $ 5 $ given distinct colors, such that each color appears the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.

In each square of an $11\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.

2008 white stones and 1 black stone are in a row. An 'action' means the following: select one black stone and change the color of neighboring stone(s). Find all possible initial position of the black stone, to make all stones black by finite actions.

There are $2019$ students sitting around circular table. Initially each of them have one candy. Teacher is allowed to pick one student, who has at least one can candy, and this student can decide, whether he gives his candy to his neighbour on the right or on the left. Prove that no matter what students teacher picks during the process, students can always ensure that any point of time no student has more than $2$ candies.

Sasha and Serg plays next game with $100$-angled regular polygon . In the beggining Sasha set natural numbers in every angle. Then they make turn by turn, first turn is made by Serg. Serg turn is to take two opposite angles and add $1$ to its numbers. Sasha turn is to take two neigbour angles and add $1$ to its numbers. Serg want to maximize amount of odd numbers. What maximal number of odd numbers can he get no matter how Sasha plays?

Let $m$ and $n$ be fixed distinct positive integers. A wren is on an infinite board indexed by $\mathbb Z^2$, and from a square $(x,y)$ may move to any of the eight squares $(x\pm m, y\pm n)$ or $(x\pm n, y \pm m)$. For each $\{m,n\}$, determine the smallest number $k$ of moves required to travel from $(0,0)$ to $(1,0)$, or prove that no such $k$ exists. [i]Proposed by Michael Ren

Let $M = \{1, 2, 3, 4, 6, 8,12,16, 24, 48\} .$ Find out which of four-element subsets of $M$ are more: those with product of all elements greater than $2009$ or those with product of all elements less than $2009.$

Consider a finite binary string $b$ with at least $2017$ ones. Show that one can insert some plus signs in between pairs of digits such that the resulting sum, when performed in base $2$, is equal to a power of two. [i]Proposed by David Stoner