Problem 5

Let

n

be a positive integer. A pair of

n

-tuples

(a_1,\ldots,a_n)

and

(b_1,\ldots,b_n)

with integer entries is called an exquisite pair if

|a_1b_1+\cdots+a_nb_n|\le1

. Determine the maximum number of distinct

n

-tuples with integer entries such that any two of them form an exquisite pair.

Step 2 of 5: Verify the exquisite condition

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\text{at most two coordinates of } a_ib_i \text{ are nonzero}; \text{ same pair, opposite signs} \implies \text{sum} \in\{-1,0,1\}

Detailed analysis

For any two tuples in this list, each has at most two nonzero entries, so at most two of the products $a_ib_i$ can be nonzero. The only way two products are simultaneously nonzero is when both tuples are supported on the same pair of positions $i,j$ : comparing $(1,1)$ with $(1,-1)$ gives dot product $1\cdot1+1\cdot(-1)=0$ , and any single-entry or zero tuple paired with another contributes at most one nonzero product of size $1$ . Hence every pairwise dot product lies in $\{-1,0,1\}$ , so all pairs are exquisite.