Question 114
Let $n \ge 2$ be a natural number. The sets $A_1, \dots, A_n$ and $B_1, \dots, B_n$ of natural numbers satisfy the properties:
* $A_i \cap B_j \neq \emptyset$ for all $i, j \in \{1, 2, \dots, n\}$;
* $A_i \cap A_j = \emptyset$ and $B_i \cap B_j = \emptyset$ for all $i \neq j \in \{1, 2, \dots, n\}$.
For each of the sets we arrange its elements in descending order and compute the largest difference between two adjacent elements in the result ordering. Find the smallest possible value of the largest among these differences.
* $A_i \cap B_j \neq \emptyset$ for all $i, j \in \{1, 2, \dots, n\}$;
* $A_i \cap A_j = \emptyset$ and $B_i \cap B_j = \emptyset$ for all $i \neq j \in \{1, 2, \dots, n\}$.
For each of the sets we arrange its elements in descending order and compute the largest difference between two adjacent elements in the result ordering. Find the smallest possible value of the largest among these differences.