How many ways are there to completely fill a $3 \times 3$ grid of unit squares with the letters $B, M$, and $T$, assigning exactly one of the three letters to each of the squares, such that no $2$ adjacent unit squares contain the same letter? Two unit squares are adjacent if they share a side.

Carson the farmer has a plot of land full of crops in the shape of a $6 \times 6$ grid of squares. Each day, he uniformly at random chooses a row or a column of the plot that he hasn’t chosen before and harvests all of the remaining crops in the row or column. Compute the expected number of connected components that the remaining crops form after $6$ days. If all crops have been harvested, we say there are $0$ connected components.

A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ .

New license plates consist of two letters, three digits, and two letters (from the English alphabet of$ 26$ letters). What is the largest possible number of such license plates if it is required that every two of them differ at no less than two positions?

We are given $1999$ coins. No two coins have the same weight. A machine is provided which allows us with one operation to determine, for any three coins, which one has the middle weight. Prove that the coin that is the $1000$th by weight can be determined using no more than $1000000$ operations and that this is the only coin whose position by weight can be determined using this machine.

There's a group of $21$ people such that each person has no more than $7$ friends among the others and any two friends have a different number of total friends. Prove that there are $6$ people, none of which knows the others.

Let $n$ be a natural number. A tiling of a $2n \times 2n$ board is a placing of $2n^2$ dominos (of size $2 \times 1$ or $1 \times 2$) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two [i]sepearate tilings[/i] of a $2n \times 2n$ board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours. Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that $n$ is divisible by $3$ [i] Proposed by Tejaswi Navilarekallu [/i]

Let $ X$ be a set containing $ 2k$ elements, $ F$ is a set of subsets of $ X$ consisting of certain $ k$ elements such that any one subset of $ X$ consisting of $ k \minus{} 1$ elements is exactly contained in an element of $ F.$ Show that $ k \plus{} 1$ is a prime number.

Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors.

How many integers $n$ from $1$ to $2020$, inclusive, are there such that $2020$ divides $n^2 + 1$?

Two players $A$ and $B$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $A$, $B$, $A$, $....$, $A$ starts the game and the one who takes out the last stone loses.$ B$ can serve on each play $1$, $2$ or 3 stones, while$ A$ can draw $2, 3, 4$ stones or $1$ stone in each turn f it is the last one in the pile. Determine for what values of $N$ does $A$ have a winning strategy, and for what values the winning strategy is $B$'s.

Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.

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 $m+n$.

Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions: $\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$ $\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and $\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$ (Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)

Consider positive integers $2,3,\ldots,n-1,n$ where $n\ge96.$ Consider any partition in two (sub)sets. Show that at least one of these two sets always contains two numbers and their product. Show that the statement does not hold for $n=95,$ e.g. there is a partition without the mentioned property.

Every single point on the plane with integer coordinates is coloured either red, green or blue. Find the least possible positive integer $n$ with the following property: no matter how the points are coloured, there is always a triangle with area $n$ that has its $3$ vertices with the same colour.

The Wonder Island is inhabited by Hedgehogs. Each Hedgehog consists of three segments of unit length having a common endpoint, with all three angles between them $120^{\circ}$. Given that all Hedgehogs are lying flat on the island and no two of them touch each other, prove that there is a finite number of Hedgehogs on Wonder Island.

Let ${a_1, a_2, . . . , a_n,}$ and ${b_1, b_2, . . . , b_n}$ be real numbers with ${a_1, a_2, . . . , a_n}$ distinct. Show that if the product ${(a_i + b_1)(a_i + b_2) \cdot \cdot \cdot (a_i + b_n)}$ takes the same value for every ${ i = 1, 2, . . . , n, }$ , then the product ${(a_1 + b_j)(a_2 + b_j) \cdot \cdot \cdot (a_n + b_j)}$ also takes the same value for every ${j = 1, 2, . . . , n, }$ .

During the mathematics Olympiad, students solved three problems. Each task was evaluated with an integer number of points from $0$ to $7$. There is at most one problem for each pair of students, for which they got after the same number of points. Determine the maximum number of students could participate in the Olympics.

Five children are divided into groups and in each group they take the hand forming a wheel to dance spinning. How many different wheels those children can form, if it is valid that there are groups of $1$ to $5$ children, and can there be any number of groups?

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Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?

There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other. (Two rooks are not threatening each other if there is a block lying between them.)

Given an integer $n\ge2$, the graph $G$ is defined by: - Vertices of $G$ are represented by binary strings of length $n$ - Two vertices $a,b$ are connected by an edge if and only if they differ in exactly $2$ places Let $S$ be a subset of the vertices of $G$, and let $S'$ be the set of edges between vertices in $S$ and vertices not in $S$. Show that if $|S|$ (the size of $S$) $\le2^{n-2}$, then $|S'|\ge|S|$.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.