To celebrate the 20th Thailand Mathematical Olympiad (TMO), Ratchasima Witthayalai School put up flags around the Thao Suranari Monument so that [list=i] [*] Each flag is painted in exactly one color, and at least $2$ distinct colors are used. [*] The number of flags are odd. [*] Every flags are on a regular polygon such that each vertex has one flag. [*] Every flags with the same color are on a regular polygon. [/list] Prove that there are at least $3$ colors with the same amount of flags.

Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A grid consists of all points of the form $(m, n)$ where $m$ and $n$ are integers with $|m|\le 2019,|n| \le 2019$ and $|m| +|n| < 4038$. We call the points $(m,n)$ of the grid with either $|m| = 2019$ or $|n| = 2019$ the [i]boundary points[/i]. The four lines $x = \pm 2019$ and $y= \pm 2019$ are called [i]boundary lines[/i]. Two points in the grid are called [i]neighbours [/i] if the distance between them is equal to $1$. Anna and Bob play a game on this grid. Anna starts with a token at the point $(0,0)$. They take turns, with Bob playing first. 1) On each of his turns. Bob [i]deletes [/i] at most two boundary points on each boundary line. 2) On each of her turns. Anna makes exactly three [i]steps[/i] , where a [i]step [/i] consists of moving her token from its current point to any neighbouring point, which has not been deleted. As soon as Anna places her token on some boundary point which has not been deleted, the game is over and Anna wins. Does Anna have a winning strategy? [i]Proposed by Demetres Christofides, Cyprus[/i]

One hundred points are marked in the plane, with no three in a line. Is it always possible to connect the points in pairs such that all fifty segments intersect one another?

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

We call a sequence of $n$ digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code $(1101)$ contains $10$ palindromes, of which $6$ are nice.) a) What is the least number of palindromes in a code? b) What is the least number of nice palindromes in a code?

Let $n$ be an integer greater than $1.$ $n$ pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of $n.$ They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.

Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that - One person should book at most one admission ticket for one match; - At most one match was same in the tickets booked by every two persons; - There was one person who booked six tickets. How many tickets did those football fans book at most?

The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this: [img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img] The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .

Four coins are laid out on a table so that they form the corners of a square. One move consists of tipping one of the coins by letting it jump over one of the others the coin so that it ends up on the directly opposite side of the other coin, the same distance from as it was before the move was made. Is it possible to make a number of moves so that the coins ends up in the corners of a square with a different side length than the original square?

Find the number of ordered pairs $(a, b, c, d)$ of positive integers satisfying the equation: \[a + 2b + 3c + 1000d = 2024.\] [i]Proposed by Irakli Khutsishvili, Georgia [/i]

In the space, there is a convex polyhedron $D$ such that for every vertex of $D$, there are an even number of edges passing through that vertex. We choose a face $F$ of $D$. Then we assign each edge of $D$ a positive integer such that for all faces of $D$ different from $F$, the sum of the numbers assigned on the edges of that face is a positive integer divisible by $2024$. Prove that the sum of the numbers assigned on the edges of $F$ is also a positive integer divisible by $2024$.

Let $n> 2$ be a natural. Given $2n$ points in the plane, no $3$ are collinear. What is the maximum number of lines that can be drawn between them, without forming a triangle? [hide=original wording]Sea n > 2 un natural. Dados 2n puntos en el plano, tres a tres no colineales, Cual es el numero maximo de trazos que pueden dibujarse entre ellos, sin formar un triangulo?[/hide]

We say that a rectangle with side lengths $a$ and $b$ [i]fits inside[/i] a rectangle with side lengths $c$ and $d$ if either ($a \le c$ and $b \le d$) or ($a \le d$ and $b \le c$). For instance, a rectangle with side lengths $1$ and $5$ [i]fits inside[/i] another rectangle with side lengths $1$ and $5$, and also [i]fits inside[/i] a rectangle with side lengths $6$ and $2$. Suppose $S$ is a set of $2019$ rectangles, all with integer side lengths between $1$ and $2018$ inclusive. Show that there are three rectangles $A$, $B$, and $C$ in $S$ such that $A$ fits inside $B$, and $B$ [i]fits inside [/i]$C$.

For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$

Let $m_1, m_2, \ldots, m_n$ be a collection of $n$ positive integers, not necessarily distinct. For any sequence of integers $A = (a_1, \ldots, a_n)$ and any permutation $w = w_1, \ldots, w_n$ of $m_1, \ldots, m_n$, define an [i]$A$-inversion[/i] of $w$ to be a pair of entries $w_i, w_j$ with $i < j$ for which one of the following conditions holds: [list] [*]$a_i \ge w_i > w_j$ [*]$w_j > a_i \ge w_i$, or [*]$w_i > w_j > a_i$. [/list] Show that, for any two sequences of integers $A = (a_1, \ldots, a_n)$ and $B = (b_1, \ldots, b_n)$, and for any positive integer $k$, the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $A$-inversions is equal to the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $B$-inversions.

$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition: (1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$ (2) $d \mid (x_1+x_2+ \cdots x_n)$ Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.

In a social network with a fixed finite setback of users, each user had a fixed set of [i]followers[/i] among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight. Let $m$ be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days. [i]Vladislav Novikov[/i]

Let $n \ge 3$ be an integer and $S$ be a set of $n$ elements. Determine the largest integer $k_n$ such that: for each selection of $k_n$ $3-$subsets of $S$, there exists a way to color elements of $S$ with two colors such that none of the chosen $3-$subset is monochromatic.

Consider a board on $2013 \times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other?

In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$. We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle. The following property is satisfied: "for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element" Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends. ([i]Serbia[/i])

There are $n$ points on a flat piece of paper, any two of them at a distance of at least $2$ from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals $\frac 32$. Prove that there exist two vectors of equal length less than $1$ and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.

Given a graph with $n$ ($n\ge 4$) vertices . It is known that for any two vertices $A$ and $B$ there exists a vertex which is connected by edges both with $A$ and $B$. Find the smallest possible numbers of edges in the graph. E. Barabanov

Fibonacci numbers are defined as follows: $F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0$. Let $a_n$ be the number of words that consist of $n$ letters $0$ or $1$ and contain no two letters $1$ at distance two from each other. Express $a_n$ in terms of Fibonacci numbers.

A certain frog that was placed on a vertex of a convex polygon chose to jump to another vertex, either clockwise skipping one vertex, either counterclockwise skipping two vertexes, and repeated the procedure. If the number of jumps that the frog made is equal to the number of sides of the polygon, the frog has passed through all its vertexes and ended up on the initial vertex, what´s the set formed by all the possible values that this number can take? [i]Andrei Eckstein[/i]