Let $ A_1A_2...A_n$ be a convex polygon. Show that there exists an index $ j$ such that the circum-circle of the triangle $ A_j A_{j \plus{} 1} A_{j \plus{} 2}$ covers the polygon (here indices are read modulo n).

There are $12$ delegates in a mathematical conference. It is known that every two delegates share a common friend. Prove that there is a delegate who has at least five friends in that conference. [i]Proposed by Nairy Sedrakyan[/i]

Consider a table with one real number in each cell. In one step, one may switch the sign of the numbers in one row or one column simultaneously. Prove that one can obtain a table with non-negative sums in each row and each column.

William was bored at the math lesson, so he drew a circle and $n\ge3$ empty cells around the circumference. In every cell he wrote a positive number. Later on he erased the numbers and in every cell wrote the geometric mean of the numbers previously written in the two neighboring cells. Show that there exists a cell whose number was not replaced by a larger number.

We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ [i]nice [/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element [i]nice [/i] subset while keeping it nice, find the minimum value of $n$.

There are $2015$ distinct circles in a plane, with radius $1$. Prove that you can select $27$ circles, which form a set $C$, which satisfy the following. For two arbitrary circles in $C$, they intersect with each other or For two arbitrary circles in $C$, they don't intersect with each other.

Can the number obtained by writing the numbers from 1 to $n$ in order ($n > 1$) be the same when read left-to-right and right-to-left? [i]N. Agakhanov[/i]

Find all finite sets of positive integers with at least two elements such that for any two numbers $ a$, $ b$ ($ a > b$) belonging to the set, the number $ \frac {b^2}{a \minus{} b}$ belongs to the set, too.

In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$. We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle. The following property is satisfied: "for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element" Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends. ([i]Serbia[/i])

There are $2015$ distinct circles in a plane, with radius $1$. Prove that you can select $27$ circles, which form a set $C$, which satisfy the following. For two arbitrary circles in $C$, they intersect with each other or For two arbitrary circles in $C$, they don't intersect with each other.

We are given a convex polygon. Show that one can find a point $Q$ inside the polygon and three vertices $A_1,A_2,A_3$ (not necessarily consecutive) such that each ray $A_iQ$ ($i=1,2,3$) makes acute angles with the two sides emanating from $A_i$.

Let $S$ be a set of positive integers such that if $a$ and $b$ are elements of $S$ such that $a<b$, then $b-a$ divides the least common multiple of $a$ and $b$, and the quotient is an element of $S$. Prove that the cardinality of $S$ is less than or equal to $2$.

$n$ red and $n$ blue points on a plane are given so that no three of the $2n$ points are collinear. Prove that it is always possible to split up the points into $n$ pairs, with one red and one blue point in each pair, so that no two of the $n$ line segments which connect the two members of a pair intersect.

There are 999 scientists. Every 2 scientists are both interested in exactly 1 topic and for each topic there are exactly 3 scientists that are interested in that topic. Prove that it is possible to choose 250 topics such that every scientist is interested in at most 1 theme. [i]A. Magazinov[/i]

In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$. We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle. The following property is satisfied: "for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element" Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends. ([i]Serbia[/i])