OBM 2018

We say that a polygon $P$ is inscribed in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$ . We also say in this case that $Q$ is circumscribed to $P$ . Given a triangle $T$ , let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$ . Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs.

Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$ ; or by $q-1$ ; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$ . The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations.a) Show that if the initial number written is $0$ , then Azambuja cannot reach his goal.b) Find all initial numbers for which Azambuja can achieve his goal.

Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$ , all belonging to the interval $[0, 1]$ , around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$ , $p_2 = x_2x_3$ , $\dots$ , $p_{2n} = x_{2n}x_1$ . She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?