Question 50
Let $n \ge 2$ be an integer, and let $z_1, \dots, z_n$ be positive integers satisfying:
* $z_j \le j$ for $j = 1, \dots, n$;
* $z_1 + \dots + z_n$ is even.
Prove that there exist $s_1, \dots, s_n \in \{-1, 1\}$ such that:
$$
s_1z_1 + s_2z_2 + \dots + s_nz_n = 0.
$$
* $z_j \le j$ for $j = 1, \dots, n$;
* $z_1 + \dots + z_n$ is even.
Prove that there exist $s_1, \dots, s_n \in \{-1, 1\}$ such that:
$$
s_1z_1 + s_2z_2 + \dots + s_nz_n = 0.
$$