KMO 2001 Problem 6

For a positive integer $n \ge 5$ , let $a_i,b_i (i = 1,2, \cdots ,n)$ be integers satisfying the following two conditions:

[list] (a) The pairs $(a_i,b_i)$ are distinct for $i = 1,2,\cdots,n$ ; (b) $|a_1b_2-a_2b_1| = |a_2b_3-a_3b_2| = \cdots = |a_nb_1-a_1b_n| = 1$ . [/list]

Prove that there exist indices $i,j$ such that $1<|i-j|<n-1$ and $|a_ib_j-a_jb_i|=1$ .