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APMO 2017: Exquisite pairs

Source: APMO 2017, problem 5

May 14, 2017
APMOcombinatorics

Problem Statement

Let nn be a positive integer. A pair of nn-tuples (a1,⋯,an)(a_1,\cdots{}, a_n) and (b1,⋯,bn)(b_1,\cdots{}, b_n) with integer entries is called an exquisite pair if ∣a1b1+⋯+anbn∣≤1.|a_1b_1+\cdots{}+a_nb_n|\le 1. Determine the maximum number of distinct nn-tuples with integer entries such that any two of them form an exquisite pair.
Pakawut Jiradilok and Warut Suksompong, Thailand
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