A valid lottery ticket is formed by choosing $5$ distinct numbers from $1, 2,3,..., 90$. What is the probability that the winning ticket contains at least two consecutive numbers?

In how many ways we can paint a $N \times N$ chessboard using 4 colours such that squares with a common side are painted with distinct colors and every $2 \times 2$ square (formed with 4 squares in consecutive lines and columns) is painted with the four colors?

In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$, $ n\in\mathbb{Z}$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color. [b]IMO Shortlist 2007 Problem C5 as it appears in the official booklet:[/b] In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$ for every integer $ n.$ Assume each strip $ S_n$ is colored either red or blue, and let $ a$ and $ b$ be two distinct positive integers. Prove that there exists a rectangle with side length $ a$ and $ b$ such that its vertices have the same color. ([i]Edited by Orlando Döhring[/i]) [i]Author: Radu Gologan and Dan Schwarz, Romania[/i]

Let $n$ be a positive integer and let $G_n$ be the set of all simple graphs on $n$ vertices. For each vertex $v$ of a graph in $G_n$, let $k(v)$ be the maximal cardinality of an independent set of neighbours of $v$. Determine $max_{G \in G_n} \Sigma_{v\in V (G)}k(v)$ and the graphs in $G_n$ that achieve this value.

Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.

Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set $$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$ Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, x_3, y_1, y_2, y_3$ such that $x_1 < x_2 < x_3, y_1 < y_2 < y_3$ and $$(x_1, y_2), (x_2, y_1), (x_2, y_2), (x_2, y_3), (x_3, y_2) \in A.$$ Determine the largest possible number of elements in $A$.

Let $n$ be a positive integer. There are $3n$ women's volleyball teams in the tournament, with no more than one match between every two teams (there are no ties in volleyball). We know that there are $3n^2$ games played in this tournament. Proof: There exists a team with at least $\frac{n}{4}$ win and $\frac{n}{4}$ loss

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.

Let $A$ be a set of positive integers having the following property: for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$. Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.

For a positive integer $n$, a [i]sum-friendly odd partition[/i] of $n$ is a sequence $(a_1, a_2, \ldots, a_k)$ of odd positive integers with $a_1 \le a_2 \le \cdots \le a_k$ and $a_1 + a_2 + \cdots + a_k = n$ such that for all positive integers $m \le n$, $m$ can be [b]uniquely[/b] written as a subsum $m = a_{i_1} + a_{i_2} + \cdots + a_{i_r}$. (Two subsums $a_{i_1} + a_{i_2} + \cdots + a_{i_r}$ and $a_{j_1} + a_{j_2} + \cdots + a_{j_s}$ with $i_1 < i_2 < \cdots < i_r$ and $j_1 < j_2 < \cdots < j_s$ are considered the same if $r = s$ and $a_{i_l} = a_{j_l}$ for $1 \le l \le r$.) For example, $(1, 1, 3, 3)$ is a sum-friendly odd partition of $8$. Find the number of sum-friendly odd partitions of $9999$.

Esteban the alchemist have $8088$ copper pieces, $6066$ bronze pieces, $4044$ silver pieces and $2022$ gold pieces. He can take two pieces of different metals and use a magic hammer to turn them into two pieces of different metals that he take and different each other. Find the largest number of gold pieces that Esteban can obtain after using the magic hammer a finite number of times. $\textbf{Note:}$ [i]If Esteban takes a copper and bronze pieces, then he turn them into a silver and a gold pieces.[/i]

Consider $h+1$ chess boards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of $h?$

Let $n$ and $m$ be positive integers. The daycare nanny uses $n \times m$ square floor mats to construct an $n \times m$ rectangular area, with a baby on each of the mats. Each baby initially faces toward one side of the rectangle. When the nanny claps, all babies crawl one mat forward in the direction it is facing at, and then turn 90 degrees clockwise. If a baby crawls outside of the rectangle, it cries. If two babies simultaneously crawl onto the same mat, they bump into each other and cry. Suppose that it is possible for the nanny to arrange the initial direction of each baby so that, no matter how many times she claps, no baby would cry. Find all possible values of $n$ and $m$. [i]Proposed by Chu-Lan Kao[/i]

There is a shape like this (Attachment down below) Find the number of triangles made by choosing 3 vertex from the 8 vertex in the attachment.

Suppose that we have $ n \ge 3$ distinct colours. Let $ f(n)$ be the greatest integer with the property that every side and every diagonal of a convex polygon with $ f(n)$ vertices can be coloured with one of $ n$ colours in the following way: (i) At least two colours are used, (ii) any three vertices of the polygon determine either three segments of the same colour or of three different colours. Show that $ f(n) \le (n\minus{}1)^2$ with equality for infintely many values of $ n$.

A circle is divided into $ n $ equal parts. Marceline sets out to assign whole numbers from $ 1 $ to $ n $ to each of these pieces so that the distance between two consecutive numbers is always the same. The numbers $ 887 $, $ 217 $ and $ 1556 $ occupy consecutive positions. How many parts was the circumference divided into?

Is it true that in any convex $n$-gon with $n > 3$, there exists a vertex and a diagonal passing through this vertex such that the angles of this diagonal with both sides adjacent to this vertex are acute? [i]Proposed by Boris Frenkin - Russia[/i]

$n$ coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations?

We have a table in the form of a regular polygon with $1000$ sides, where each side has length $1$. At one of the vertices is a beetle (consider this vertex to be fixed). The $1000$ vertices must be numbered, in some order, using the numbers $1, 2,\ldots ,1000$ such that the beetle is at vertex $1$. The beetle can only move along the edge of the table and always moves clockwise. The beetle moves from vertex $1$ to vertex $2$ and stops there. then it moves from vertex $2$ to vertex $3$, and stops there. So on, until the beetle ends its journey at vertex $1000$. Find the number of ways the numbers can be assigned to the vertices so that the total length of the beetle's journey is $2017$.

Connecting the vertices of a regular $ n$-gon we obtain a closed (not necessarily convex) $ n$-gon. Show that if $ n$ is even, then there are two parallel segments among the connecting segments and if $ n$ is odd then there cannot be exactly two parallel segments.

5. $A$ and $B$ play alternating turns on a $2012 \times 2013$ board with enough pieces of the following types: Type $1$: Piece like Type $2$ but with one square at the right of the bottom square. Type $2$: Piece of $2$ consecutive squares, one over another. Type $3$: Piece of $1$ square. At his turn, $A$ must put a piece of the type $1$ on available squares of the board. $B$, at his turn, must put exactly one piece of each type on available squares of the board. The player that cannot do more movements loses. If $A$ starts playing, decide who has a winning strategy. Note: The pieces can be rotated but cannot overlap; they cannot be out of the board. The pieces of the types $1$, $2$ and $3$ can be put on exactly $3$, $2$ and $1$ squares of the board respectively.

Find all the pairs of integers $ (a,b)$ satisfying $ ab(a \minus{} b)\not \equal{} 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n \plus{} a,n \plus{} b$ belongs to $ Z_{0}$.

There are $1000$ rooms in a row along a long corridor. Initially the first room contains $1000$ people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?

A permutation of the integers $1, 2, \ldots, m$ is called [i]fresh[/i] if there exists no positive integer $k < m$ such that the first $k$ numbers in the permutation are $1, 2, \ldots, k$ in some order. Let $f_m$ be the number of fresh permutations of the integers $1, 2, \ldots, m$. Prove that $f_n \ge n \cdot f_{n - 1}$ for all $n \ge 3$. [i]For example, if $m = 4$, then the permutation $(3, 1, 4, 2)$ is fresh, whereas the permutation $(2, 3, 1, 4)$ is not.[/i]

For each letter in the English alphabet, William assigns an English word which contains that letter. His first document consists only of the word assigned to the letter $A$. In each subsequent document, he replaces each letter of the preceding document by its assigned word. The fortieth document begins with “Till whatsoever star that guides my moving.” Prove that this sentence reappears later in this document.