### APMO 2017 P5

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**Sat May 27, 2017 1:45 pm**Let $n$ be a positive integer. A pair of $n$-tuples $(a_1,....a_n)$ and $(b_1,...b_n)$ with integer entries is called an exquisite pair if

$|a_1b_1 +...+ a_nb_n| \leq 1$.

Determine the maximum number of distinct $n$-tuples with integer entries such that any two of them form an exquisite pair.

$|a_1b_1 +...+ a_nb_n| \leq 1$.

Determine the maximum number of distinct $n$-tuples with integer entries such that any two of them form an exquisite pair.