Question 11
Let $k$, $l$, $n$ be positive integers, and let $a_1, a_2, \dots, a_k \in \{1, 2, \dots, n\}$ satisfy the following three conditions:
(1) $n \ge 3$, $l \le n-2$, and $l-k \le \frac{n-3}{2}$;
(2) For each $t \in \{1, 2, \dots, l\}$, there exists a non-empty subset $I \subseteq \{1, 2, \dots, k\}$ such that
$$
\sum_{i \in I} a_i \equiv t \pmod{n};
$$
(3) For each $t \in \{l+1, l+2, \dots, n\}$, there does not exist a non-empty subset $I \subseteq \{1, 2, \dots, k\}$ such that
$$
\sum_{i \in I} a_i \equiv t \pmod{n}.
$$
Prove that $a_1 + a_2 + \dots + a_k = l$.
(1) $n \ge 3$, $l \le n-2$, and $l-k \le \frac{n-3}{2}$;
(2) For each $t \in \{1, 2, \dots, l\}$, there exists a non-empty subset $I \subseteq \{1, 2, \dots, k\}$ such that
$$
\sum_{i \in I} a_i \equiv t \pmod{n};
$$
(3) For each $t \in \{l+1, l+2, \dots, n\}$, there does not exist a non-empty subset $I \subseteq \{1, 2, \dots, k\}$ such that
$$
\sum_{i \in I} a_i \equiv t \pmod{n}.
$$
Prove that $a_1 + a_2 + \dots + a_k = l$.