MathDB

72

Part of 1990 IMO Longlists

Problems(1)

There exists a pair of indexes - ILL 1990 KOR1

Source:

9/18/2010
Let n≥5n \geq 5 be a positive integer. a1,b1,a2,b2,…,an,bna_1, b_1, a_2, b_2, \ldots, a_n, b_n are integers. (ai,bi)( a_i, b_i) are pairwisely distinct for i=1,2,…,ni = 1, 2, \ldots, n, and ∣a1b2−a2b1∣=∣a2b3−a3b2∣=⋯=∣an−1bn−anbn−1∣=1|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,ji, j satisfying 2≤∣i−j∣≤n−22 \leq |i - j| \leq n - 2 and ∣aibj−ajbi∣=1.|a_ib_j -a_jb_i| = 1.
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