2017 APMO 第 5 题

Let $n$ be a positive integer. A pair of $n$-tuples $\left(a_{1}, \ldots, a_{n}\right)$ and $\left(b_{1}, \ldots, b_{n}\right)$ with integer entries is called an exquisite pair if

$$\left|a_{1} b_{1}+\cdots+a_{n} b_{n}\right| \leq 1$$

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

Answer: The maximum is $n^{2}+n+1$.

令 $n$ 为正整数。一对具有整数项的 $n$ 元组 $\left(a_{1}, \ldots, a_{n}\right)$ 和 $\left(b_{1}, \ldots, b_{n}\right)$ 称为精致对，如果

$$\left|a_{1} b_{1}+\cdots+a_{n} b_{n}\right| \leq 1$$

确定具有整数条目的不同 $n$ 元组的最大数量，使得它们中的任何两个形成精美的对。

答案：最大值为$n^{2}+n+1$。