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Exponential Growth

Source: 2020 Taiwan TST Round 2 Mock Exam P5

May 25, 2020
number theory

Problem Statement

A finite set KK consists of at least 3 distinct positive integers. Suppose that KK can be partitioned into two nonempty subsets A,B∈KA,B\in K such that ab+1ab+1 is always a perfect square whenever a∈Aa\in A and b∈Bb\in B. Prove that max⁡k∈Kk≥⌊(2+3)min⁡{∣A∣,∣B∣}−1⌋+1,\max_{k\in K}k\geq \left\lfloor (2+\sqrt{3})^{\min\{|A|,|B|\}-1}\right\rfloor+1,where ∣X∣|X| stands for the cartinality of the set XX, and for x∈Rx\in \mathbb{R}, ⌊x⌋\lfloor x\rfloor is the greatest integer that does not exceed xx.
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