Ben and Connor are playing a game of wallball. The first player to lead by $2$ points wins the game. Suppose Ben wins each point with probability $\tfrac{3}{4}$ and is gracious enough to allow Connor to start with a $1$ point lead. The probability that Ben wins the game is $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ What is $m + n$?

we are given $n$ rectangles in the plane. Prove that between $4n$ right angles formed by these rectangles there are at least $[4\sqrt n]$ distinct right angles.

On a board with $a$ rows and $b$ columns, each square has a switch and an unlit light bulb. By pressing the switch of a house, the lamp in that house changes state, along with the lamps in the same row and those in the same column (those that are on go out and the that are off light up). For what values of $a$ and $b$ is it possible to have just one lamp on, by pressing a series of switches?

16 NHL teams in the first playoff round divided in pairs and to play series until 4 wins (thus the series could finish with score 4-0, 4-1, 4-2, or 4-3). After that 8 winners of the series play the second playoff round divided into 4 pairs to play series until 4 wins, and so on. After all the final round is over, it happens that $k$ teams have non-negative balance of wins (for example, the team that won in the first round with a score of 4-2 and lost in the second with a score of 4-3 fits the condition: it has $4+3=7$ wins and $2+4=6$ losses). Find the least possible $k$.

Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then \[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\] For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.

An alphabet has $n$ letters. A word is called [i]differentiated [/i] if it has the following property fulfilled: No letter occurs more than once between two identical letters. For example with the alphabet $\{a, b, c, d\}$ the word [i]abbdacbdd [/i] is not, the word [i]bbacbadcdd [/i] is differentiated. (a) Each differentiated word has a maximum of $3n$ letters. (b) How many differentiated words with exactly $3n$ letters are ther

Seven triangles of area $7$ lie in a square of area $27$. Prove that among the $7$ triangles there are $2$ that intersect in a region of area not less than $1$.

Let $S$ be a string of $99$ characters, $66$ of which are $A$ and $33$ are $B$. We call $S$ [i]good[/i] if, for each $n$ such that $1\le n \le 99$, the sub-string made from the first $n$ characters of $S$ has an odd number of distinct permutations. How many good strings are there? Which strings are good?

Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.

Consider a set $S$ of $200$ points on the plane such that $100$ points are the vertices of a convex polygon $A$ and the other $100$ points are in the interior of the polygon. Moreover, no three of the given points are collinear. A triangulation is a way to partition the interior of the polygon $A$ into triangles by drawing the edges between some two points of S such that any two edges do not intersect in the interior, and each point in $S$ is the vertex of at least one triangle. 1. Prove that the number of edges does not depend on the triangulation. 2. Show that for any triangulation, one can color each triangle by one of three given colors such that any two adjacent triangles have different colors.

Let $n$ be a positive integer. Suppose for any $\mathcal{S} \subseteq \{1, 2, \cdots, n\}$, $f(\mathcal{S})$ is the set containing all positive integers at most $n$ that have an odd number of factors in $\mathcal{S}$. How many subsets of $\{1, 2, \cdots, n\}$ can be turned into $\{1\}$ after finitely many (possibly zero) applications of $f$?

In a $2^n \times 2^n$ square with $n$ positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)

Some vertices of a regular $2024$-gon are marked such that for any regural polygon, all of whose vertices are vertices of the $2024$-gon, at least one of his vertices is marked. Find the minimal possible number of marked vertices [i]A. Voidelevich[/i]

Each lattice point of the plane is labeled by a positive integer. Each of these numbers is the arithmetic mean of its four neighbors (above, below, left, right). Show that all the numbers are equal.

A subset of a student group is called an [i]ideal company[/i] if 1) in this subset, all girls are liked by all young men, 2) no one can be added to this subset without violating condition $1$. In a certain group, $9$ female students and $15$ students study. Warden of the group made a list of all kinds of ideal companies in this group. What is the largest number of companies on this list?

In the night, stars in the sky are seen in different time intervals. Suppose for every $k$ stars ($k>1$), at least $2$ of them can be seen in one moment. Prove that we can photograph $k-1$ pictures from the sky such that each of the mentioned stars is seen in at least one of the pictures. (The number of stars is finite. Define the moments that the $n^{th}$ star is seen as $[a_n,b_n]$ that $a_n<b_n$.)

There are 12 people in a line in a bank. When the desk closes, the people form a new line at a newly opened desk. In how many ways can they do this in such a way that none of the 12 people changes his/her position in the line by more than one?

Given $2012$ stones divided into several groups, a [i]legal move[/i] is to merge two of the groups into one, as long as the size of the new group is less than or equal to $51$. Two players, $A$ and $B$, take turns making legal moves, starting with $A$. Initially, each stone is in a separate group. The player who cannot make a legal move on their turn loses. Determine which of the two players has a winning strategy and provide that strategy.

A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares?

A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices? [i]Proposed by A. Golovanov[/i]

Consider $n$ weights, $n \ge 2$, of masses $m_1, m_2, ..., m_n$, where $m_k$ are positive integers such that $1 \le m_ k \le k$ for all $k \in \{1,2,...,n\} $: Prove that we can place the weights on the two pans of a balance such that the pans stay in equilibrium if and only if the number $m_1 + m_2 + ...+ m_n$ is even. Estonian Olympiad

A square house is partitioned into an $n \times n$ grid, where each cell is a room. All neighboring rooms have a door connecting them, and each door can either be normalor inversive. If USJL walks over an inversive door, he would become inverted-USJL,and vice versa. USJL must choose a room to begin and walk pass each room exactly once. If it is inverted-USJL showing up after finishing, then he would be trapped for all eternity. Prove that USJL could always escape.

In how many ways can a $2\times n$ grid be covered by (a) 2 monominoes and $n-1$ dominoes (b) 4 monominoes and $n-2$ dominoes.

Is there a colouring of all positive integers in three colours so that for each positive integer the numbers of its divisors of any two colours differ at most by $2?$

There are weights with mass $1,3,5,....,2i+1,...$ Let $A(n)$ -is number of different sets with total mass equal $n$( For example $A(9)=2$, because we have two sets $9=9=1+3+5$). Prove that $A(n) \leq A(n+1)$ for $n>1$