Eight people are sitting at a circular table, it is known that any three consecutive people at the table have an odd number of coins (among the three people), show that each person has at least one coin.

Let $ABC$ be a triangle and let $L$, $M$, $N$ be the midpoints of the sides $BC$, $CA$ and $AB$ , respectively. The points $P$ and $Q$ lie on $AB$ and $BC$, respectively; the points $R$ and $S$ are such that $N$ is the midpoint of $PR$ and $L$ is the midpoint of $QS$. Show that if $PS$ and $QR$ are perpendicular, then their intersection lies on in the circumcircle of triangle $LMN$.

We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $.

Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.

There are $100$ stones in a pile. A partition of the heap in $k $ piles is called special if it meets that the number of stones in each pile is different and also for any partition of any of the piles into two new piles it turns out that between the $k + 1$ piles there are two that have the same number of stones (each pile contains at least one stone). a) Find the maximum number $k$, such that there is a special partition of the $100$ stones into $k $ piles. b) Find the minimum number $k $, such that there is a special partition of the $100$ stones in $k $ piles.

Given a circle $C$ and a diameter $AB$ in it, mark a point $P$ on $AB$ different from the ends. In one of the two arcs determined by $AB$ choose the points $M$ and $N$ such that $\angle APM = 60 ^ \circ = \angle BPN$. The segments $MP$ and $NP$ are drawn to obtain three curvilinear triangles; $APM $, $MPN$ and $NPB$ (the sides of the curvilinear triangle $APM$ are the segments $AP$ and $PM$ and the arc $AM$). In each curvilinear triangle a circle is inscribed, that is, a circle is built tangent to the three sides. Show that the sum of the radii of the three inscribed circles is less than or equal to the radius of $C$.