CWMO 2001 Problem 8

We call $A_1, A_2, \ldots, A_n$ an $n$ -division of $A$ if

(i) $A_1 \cup A_2 \cup \cdots \cup A_n = A$ , (ii) $A_i \cap A_j \neq \emptyset$ .

Find the smallest positive integer $m$ such that for any $14$ -division $A_1, A_2, \ldots, A_{14}$ of $A = \{1, 2, \ldots, m\}$ , there exists a set $A_i$ ( $1 \leq i \leq 14$ ) such that there are two elements $a, b$ of $A_i$ such that $b < a \leq \frac {4}{3}b$ .