Question 160
Let $a_1, a_2, \dots, a_{2023}$ be nonnegative real numbers such that $a_1 + a_2 + \dots + a_{2023} = 100$. Let $N$ denote the number of elements in the following set
$$
\{(i, j) \mid 1 \le i \le j \le 2023, a_i a_j \ge 1\}.
$$
Prove that $N \le 5050$, and determine the necessary and sufficient condition for $N = 5050$.
$$
\{(i, j) \mid 1 \le i \le j \le 2023, a_i a_j \ge 1\}.
$$
Prove that $N \le 5050$, and determine the necessary and sufficient condition for $N = 5050$.