ILL 1990 Problem 72

Let $n \geq 5$ be a positive integer. $a_1, b_1, a_2, b_2, \ldots, a_n, b_n$ are integers. $( a_i, b_i)$ are pairwisely distinct for $i = 1, 2, \ldots, n$ , and $|a_1b_2 - a_2b_1| = |a_2b_3 -a_3b_2| = \cdots = |a_{n-1}b_n -a_nb_{n-1}| = 1$ . Prove that there exists a pair of indexes $i, j$ satisfying $2 \leq |i - j| \leq n - 2$ and $|a_ib_j -a_jb_i| = 1.$