Question 26
Given an odd positive integer $n$, let $m = \frac{n+1}{2}$. Let positive integers $a_1, a_2, \dots, a_m$ be pairwise incongruent modulo $n$, and positive integers $b_1, b_2, \dots, b_m$ be pairwise incongruent modulo $n$. Prove that the set
$$
C = \{a_i + b_j \text{ (least nonnegative residue modulo } n) \mid i, j \in \{1, 2, \dots, m\}, i \neq j\}
$$
has more than $n - \sqrt{n} - \frac{1}{2}$ elements.
$$
C = \{a_i + b_j \text{ (least nonnegative residue modulo } n) \mid i, j \in \{1, 2, \dots, m\}, i \neq j\}
$$
has more than $n - \sqrt{n} - \frac{1}{2}$ elements.