Question 100
Let $A_1, A_2, \dots, A_n$ be $n$ non-empty subsets of a finite set $A$ of real numbers satisfying the following conditions:
(1) The sum of elements of $A$ is equal to $0$;
(2) Pick arbitrarily a number from each $A_i$, and their sum is strictly positive.
Prove that there exist sets $A_{i_1}, A_{i_2}, \dots, A_{i_k}$, $1 \le i_1 \lt i_2 \lt \dots \lt i_k \le n$, such that
$$ | A_{i_1} \cup A_{i_2} \cup \dots \cup A_{i_k} | \lt \frac{k}{n} | A |. $$
(1) The sum of elements of $A$ is equal to $0$;
(2) Pick arbitrarily a number from each $A_i$, and their sum is strictly positive.
Prove that there exist sets $A_{i_1}, A_{i_2}, \dots, A_{i_k}$, $1 \le i_1 \lt i_2 \lt \dots \lt i_k \le n$, such that
$$ | A_{i_1} \cup A_{i_2} \cup \dots \cup A_{i_k} | \lt \frac{k}{n} | A |. $$