Question 187
Let $n \in \mathbb{N}$, with $n \ge 2$, and let $x_1, x_2, \dots, x_n$ be positive real numbers such that $x_1 + x_2 + \dots + x_n = 1$. Define
$$
m = \min \left\{ \frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, \dots, \frac{x_n}{1+x_1+x_2+\dots+x_n} \right\}
$$
and
$$
M = \max \left\{ \frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, \dots, \frac{x_n}{1+x_1+x_2+\dots+x_n} \right\}.
$$
Determine:
a) the greatest possible value of $m$;
b) the smallest possible value of $M$.
$$
m = \min \left\{ \frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, \dots, \frac{x_n}{1+x_1+x_2+\dots+x_n} \right\}
$$
and
$$
M = \max \left\{ \frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, \dots, \frac{x_n}{1+x_1+x_2+\dots+x_n} \right\}.
$$
Determine:
a) the greatest possible value of $m$;
b) the smallest possible value of $M$.