The following figure is made up of $12$ segments and $8$ circles. As you can see, at the beginning all the circles are empty. In each operation an empty circle is chosen, it is painted red and inside it the number of red neighboring circles that the chosen circle has is written (in the first operation the chosen circle is painted red and the number $0$ is written). After $8$ operations all the circles are painted red and each one has a number written on it. Prove that, no matter how the operations are done, the sum of all the numbers at the end is the same.

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and let $G$ be the point of the segment $AD$ such that $AG = 2GD$. Let $E$ and $F$ be points on the sides $AB$ and $AC$, respectively, such that$ G$ lies on the segment $EF$. Let $M$ and $N$ be points of the segments $AE$ and $AF$, respectively, such that $ME = EB$ and $NF = FC$. a) Prove that the area of the quadrilateral $BMNC$ is equal to four times the area of the triangle $DEF$. b) Prove that the quadrilaterals $MNFE$ and $AMDN$ have the same area.

Let $R$ be the set of real numbers and $f : R \to R$ be a function that satisfies: $$f(xy) + y + f(x + f(y)) = (y + 1)f(x),$$for all real numbers $x, y$. a) Determine the value of $f(0)$. b) Prove that $f(x) = 2-x$ for every real number $x$.

For each positive integer n, the number $R(n) = 11 ... 1$ is defined, which is made up of exactly $n$ digits equal to $1$. For example, $R(5) = 11111$. Let $n > 4$ be an integer for which, by writing all the positive divisors of $R(n)$, it is true that each written digit belongs to the set $\{0, 1\}$. Show that $n$ is a power of an odd prime number. Clarification: A power of an odd prime number is a number of the form $p^a$, where $p$ is an odd prime number and $a$ is a positive integer.