IT-CESENATICO 2017

Let $a$ and $b$ be positive real numbers. Consider a regular hexagon of side $a$ , and build externally on its sides six rectangles of sides $a$ and $b$ . The new twelve vertices lie on a circle. Now repeat the same construction, but this time exchanging the roles of $a$ and $b$ ; namely; we start with a regular hexagon of side $b$ and we build externally on its sides six rectangles of sides $a$ and $b$ . The new twelve vertices lie on another circle. Show that the two circles have the same radius.

Let $n\geq 2$ be an integer. Consider the solutions of the system $$\begin{cases} n=a+b-c n=a^2+b^2-c^2 \end{cases}$$ where $a,b,c$ are integers. Show that there is at least one solution and that the solutions are finitely many.

Let $ABCD$ be a thetraedron with the following propriety: the four lines connecting a vertex and the incenter of opposite face are concurrent. Prove $AB \cdot CD= AC \cdot BD = AD\cdot BC$ .

Let $x_1 , x_2, x_3 ...$ a succession of positive integers such that for every couple of positive integers $(m,n)$ we have $x_{mn} \neq x_{m(n+1)}$ . Prove that there exists a positive integer $i$ such that $x_i \ge 2017$ .

Prove that there are infinitely many positive integers $m$ such that the number of odd distinct prime factor of $m(m+3)$ is a multiple of $3$ .