Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\tfrac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. A player wins the game if their score is higher than the other player's score. The probability Dukejukem defeats the Trollner to win WASS equals $\tfrac{m}{n}$ where $m$ and $n$ are coprime positive integers. Computer $m+n$.

A chord is drawn on a circle by choosing two points uniformly at random along its circumference. This is done two more times to obtain three total random chords. The circle is cut along these three lines, splitting it into pieces. The probability that one of the pieces is a triangle is $\frac{m}{n}$ , where $m$, $n$ are positive integers and gcd $(m,n) = 1$. Find $100m + n$.

Consider the collection of all three-element subsets drawn from the set $ \{1,2,3,4,\dots,299,300\}$. Determine the number of those subsets for which the sum of the elements is a multiple of 3.

Find all positive integer $N$ such that for any infinite triangular grid with exactly $N$ black unit equilateral triangles, there exists an equilateral triangle $S$ whose sides align with grid lines such that there is exactly one black unit equilateral triangle outside of $S.$ (ltf0501)

We have a regular $n{}$-gon, with $n\geqslant 4$. We consider the arrangements of $n{}$ numbers on its vertices, each of which is equal to 1 or 2. For each such arrangement $K{}$, we find the number of odd sums among all sums of numbers in several consecutive vertices. This number is denoted by $\alpha(K)$. [list=a] [*]Find the largest possible value of $\alpha(K)$. [*]Find the number of arrangements for which $\alpha(K)$ takes this largest possible value. [/list] [i]Proposed by P. Kozhevnikov[/i]

Two players, Red and Blue, play in alternating turns on a 10x10 board. Blue goes first. In his turn, a player picks a row or column (not chosen by any player yet) and color all its squares with his own color. If any of these squares was already colored, the new color substitutes the old one. The game ends after 20 turns, when all rows and column were chosen. Red wins if the number of red squares in the board exceeds at least by 10 the number of blue squares; otherwise Blue wins. Determine which player has a winning strategy and describe this strategy.

Seven elves are sitting at a round table. Each elf has a cup. Some cups are filled with some milk. Each elf in turn and clockwise divides all his milk between six other cups. After the seventh has done this, every cup was containing the initial amount of milk. How much milk did every cup contain, if there was three litres of milk total?

Let $a,b,c$ be three positive integers such that $a<b<c$. Consider the the sets $A,B,C$ and $X$, defined as follows: $A=\{ 1,2,\ldots ,a \}$, $B=\{a+1,a+2,\ldots,b\}$, $C=\{b+1,b+2,\ldots ,c\}$ and $X=A\cup B\cup C$. Determine, in terms of $a,b$ and $c$, the number of ways of placing the elements of $X$ in three boxes such that there are $x,y$ and $z$ elements in the first, second and third box respectively, knowing that: i) $x\le y\le z$; ii) elements of $B$ cannot be put in the first box; iii) elements of $C$ cannot be put in the third box.

A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.

Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.

$k$ is an integer, there exists a triangulation for a regular polygon with $2024$ sides and $2024$ colored dots with $k$ different colors meeting $(1)$ each color will be used at least once $(2)$ every small triangle will have at least $2$ dots that will be in the same color. Try to find the maximum value of$k$

Find the maximal number of points, such that there exist a configuration of $2023$ lines on the plane, with each lines pass at least $2$ points.

A triangle has an angle of measure $\theta$. It is dissected into several triangles. Is it possible that all angles of the resulting triangles are less than $\theta$, if (a) $\theta = 70^o$ ? (b) $\theta = 80^o$ ?

Call an $n$-tuple $(a_1, . . . , a_n)$ [i]occasionally periodic [/i] if there exist a nonnegative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+p+j}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, . . . , a_n)$ with elements from set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.

We call a matrix $\textsl{binary matrix}$ if all its entries equal to $0$ or $1$. A binary matrix is $\textsl{Good}$ if it simultaneously satisfies the following two conditions: (1) All the entries above the main diagonal (from left to right), not including the main diagonal, are equal. (2) All the entries below the main diagonal (from left to right), not including the main diagonal, are equal. Given positive integer $m$, prove that there exists a positive integer $M$, such that for any positive integer $n>M$ and a given $n \times n$ binary matrix $A_n$, we can select integers $1 \leq i_1 <i_2< \cdots < i_{n-m} \leq n$ and delete the $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th rows and $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th columns of $A_n$, then the resulting binary matrix $B_m$ is $\textsl{Good}$.

There are $100$ students who praticipate at exam.Also there are $25$ members of jury.Each student is checked by one jury.Known that every student likes $10$ jury $a)$ Prove that we can select $7$ jury such that any student likes at least one jury. $b)$ Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most $10$ students.

Is it true that from an arbitrary triangle you can cut three equal figures, the area of each of which is more than a quarter of the area triangle?

$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?

Let $n$ be a natural number. Prove that the number of all non-isosceles triangles with lengths of their sides equal to natural numbers and a perimeter $2n$ is $[\frac{n^2-6n+12}{12}]$.

Given regular $45$-gon. Can you mark its corners with the digits $\{0,1,...,9\}$ in such a way, that for every pair of digits there would be a side with both ends marked with those digits?

Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable. [i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]

The numbers $1$ through $9$ are written on a $3 \times 3$ board, without repetitions. A valid operation is to choose a row or a column of the board, and replace its three numbers $a, b, c$ (in order, i.e., the first number of the row/column is $a$, the second number of the row/column is $b$, the third number of the row/column is $c$) with either the three non-negative numbers $a-x, b-x, c+x$ (in order) or with the three non-negative numbers $a+x, b-x, c-x$ (in order), where $x$ is a real positive number which may vary in each operation . a) Determine if there is a way of getting all $9$ numbers on the board to be the same, starting with the following board: $\begin{array}{|c|c|c|c|c|c|c|c|} \hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end{array}$ b) For all posible configurations such that it is possible to get all $9$ numbers to be equal to a number $m$ using the valid operations, determine the maximum value of $m$.

In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.

In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is [i]nice[/i] if it has at least one of the vertices of the $n$ rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$. (There can be nonconvex regions as well as regions with more than one boundary curve.)

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.