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ab+1 is a perfect square

Source: 2012 China TST Test 3 p4

March 26, 2012
logarithmsfloor functionfunctionnumber theory proposednumber theory

Problem Statement

Given an integer n≥4n\ge 4. S={1,2,…,n}S=\{1,2,\ldots,n\}. A,BA,B are two subsets of SS such that for every pair of (a,b),a∈A,b∈B,ab+1(a,b),a\in A,b\in B, ab+1 is a perfect square. Prove that min⁡{∣A∣,∣B∣}≤log⁡2n.\min \{|A|,|B|\}\le\log _2n.
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