During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.

Six points are given so that there are not three on the same line and that the lengths of the segments determined by these points are all different. We consider all the triangles that they have their vertices at these points. Show that there is a segment that is both the shortest side of one of those triangles and the longest side of another.

Let $S$ be a finite set of cells in a square grid. On each cell of $S$ we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of $S$ still contains one grasshopper. [list] [*] Prove that, for every set $S$, the number of Asturian arrangements is a perfect square. [*] Compute the number of Asturian arrangements if $S$ is the following set:

Let $ G$ be a graph with $ 2n$ vertexes and $ 2n(n\minus{}1)$ edges.If we color some edge to red,then vertexes,which are connected by this edge,must be colored to red too. But not necessary that all edges from the red vertex are red. Prove that it is possible to color some vertexes and edges in $ G$,such that all red vertexes has exactly $ n$ red edges.

Consider a 3 × 4 rectangular table where each cell can be colored using one of three available colors. Determine the number of different ways the table can be colored such that no two cells sharing a common side have the same color. It is not necessary to use all three colors in each coloring. [i]Proposed by Prudencio Guerrero Fernández, Cuba[/i]

Let $S$ be a $k$-element set. [i](a)[/i] Find the number of mappings $f : S \to S$ such that \[\text{(i) } f(x) \neq x \text{ for } x \in S, \quad \text{(ii) } f(f(x)) = x \text{ for }x \in S.\] [i](b)[/i] The same with the condition $\text{(i)}$ left out.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Three triangles (blue, green and red) have a common interior point $M$. Prove that it is possible to choose one vertex from each triangle so that point $M$ is located inside the triangle formed by these selected vertices. (Imre Barani, Hungary)

The set of $2000$-digit integers are divided into two sets: the set $M$ consisting all integers each of which can be represented as the product of two $1000$-digit integers, and the set $N$ which contains the other integers. Which of the sets $M$ and $N$ contains more elements?

Given a convex pentagon $ABCDE$, prove that if an arbitrary point $M$ inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass. Note: In reality, we need to exclude the points of the diagonals, because that in this case the drawn lines can pass not through the internal points of the sides, but through the vertices. But if the drawn diagonals are not considered or counted twice (because they are drawn from two vertices), then the statement remains true.

Consider all words of length $n$ consisting of the letters $I,O,M$. How many such words are there, which contain no two consecutive $M$’s?

A subset $M$ of $\{1, 2, . . . , 2006\}$ has the property that for any three elements $x, y, z$ of $M$ with $x < y < z$, $x+ y$ does not divide $z$. Determine the largest possible size of $M$.

Let $n,k$ be positive integers, satisfying $n$ is even, $k\geq 2$ and $n>4k.$ There are $n$ points on the circumference of a circle. If the endpoints of $\frac{n}{2}$ chords in a circle that do not intersect with each other are exactly the $n$ points, we call these chords a matching.Determine the maximum of integer $m,$ such that for any matching, there exists $k$ consecutive points, satisfying all the endpoints of at least $m$ chords are in the $k$ points.

The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?

$N$ straight lines are drawn on a plane. The $N$ lines can be partitioned into set of lines such that if a line $l$ belongs to a partition set then all lines parallel to $l$ make up the rest of that set. For each $n>=1$,let $a_n$ denote the number of partition sets of size $n$. Now that $N$ lines intersect at certain points on the plane. For each $n>=2$ let $b_n$ denote the number of points that are intersection of exactly $n$ lines. Show that $\sum_{n>= 2}(a_n+b_n)$$\binom{n}{2}$ $=$ $\binom{N}{2}$

Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.

An ant walks on the plane as follows: initially, it walks $1$ cm in any direction. After, at each step, it changes the trajectory direction by $60^o$ left or right and walks $1$ cm in that direction. It is possible that it returns to the point from which it started in (a) $2008$ steps? (b) $2009$ steps? [img]https://cdn.artofproblemsolving.com/attachments/8/b/d4c0d03c67432c4e790b465a74a876b938244c.png[/img]

Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that: (i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again, (ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps, (iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.

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.

There are four boats on one of the river banks; their names are Eight, Four, Two and One, because that is the number of hours it takes each of them to cross the river. One boat can be tied to another, but not more than one, and then the time it takes to cross is equal to that of the slower of the two boats. A single sailor must take all the boats to the other shore. What is the least amount of time you need to complete the move?

a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying: [b]i.[/b] each has $k$ elements; [b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$) [b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements. b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.

Consider a string of $n$ $1's$. We wish to place some $+$ signs in between so that the sum is $1000$. For instance, if $n=190$, one may put $+$ signs so as to get $11$ ninety times and $1$ ten times , and get the sum $1000$. If $a$ is the number of positive integers $n$ for which it is possible to place $+$ signs so as to get the sum $1000$, then find the sum of digits of $a$.

Each of the cells of an $m \times n$ table is coloured either black or white. For each cell, the total number of the cells which are in the same row or in the same column and of the same colour as this cell is strictly less than the total number of the cells which are in the same row or in the same column and of the other colour as this cell. Prove that in each row and in each column the number of white cells is the same as the number of black ones. (A Shapovalov)

Let there be $n$ students, numbered $1$ through $n$. Let there be $n$ cards with numbers $1$ through $n$ written on them. Each student picks a card from the stack, and two students are called a pair if they pick each other's number. Let the probability that there are no pairs be $p_n$. Prove that $p_n - p_{n-1}=0$ if $n$ is odd, and prove that $p_n - p_{n-1}= \frac{1}{(-2)^kk^{1-k}}$ if $n = 2k$.