Question 40
Let $n \ge 3$ be fixed. A sequence $a_1, a_2, \dots, a_n$ of real numbers is *nice* if
$$
0 \le a_1 + \dots + a_{k-1} + a_{k+1} + \dots + a_n \le 1
$$
for all $1 \le k \le n$. Let $m = \min\{a_1, a_2, \dots, a_n\}$ denote the minimum and let $M = \max\{a_1, a_2, \dots, a_n\}$ denote the maximum of the sequence $a_1, a_2, \dots, a_n$.
(i) Find the maximum of $M$ over all nice sequences.
(ii) Find the minimum of $m$ over all nice sequences.
$$
0 \le a_1 + \dots + a_{k-1} + a_{k+1} + \dots + a_n \le 1
$$
for all $1 \le k \le n$. Let $m = \min\{a_1, a_2, \dots, a_n\}$ denote the minimum and let $M = \max\{a_1, a_2, \dots, a_n\}$ denote the maximum of the sequence $a_1, a_2, \dots, a_n$.
(i) Find the maximum of $M$ over all nice sequences.
(ii) Find the minimum of $m$ over all nice sequences.