IT-CESENATICO 2021 Problem 4

Given two fractions $a/b$ and $c/d$ we define their pirate sum as: $\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}$ where the two initial fractions are simplified the most possible, like the result. For example, the pirate sum of $2/7$ and $4/5$ is $1/2$ . Given an integer $n \ge 3$ , initially on a blackboard there are the fractions: $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}$ . At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction. Determine, in function of $n$ , the maximum and the minimum possible value for the last fraction.