2006 APMO 第 1 题

Let $n$ be a positive integer. Find the largest nonnegative real number $f(n)$ (depending on $n$ ) with the following property: whenever $a_{1}, a_{2}, \ldots, a_{n}$ are real numbers such that $a_{1}+a_{2}+\cdots+a_{n}$ is an integer, there exists some $i$ such that $\left|a_{i}-\frac{1}{2}\right| \geq f(n)$.

令 $n$ 为正整数。找到最大的非负实数 $f(n)$ (取决于 $n$ )，其性质如下：只要 $a_{1}, a_{2}, \ldots, a_{n}$ 是实数，使得 $a_{1}+a_{2}+\cdots+a_{n}$ 是整数，则存在一些 $i$ 使得 $\left|a_{i}-\frac{1}{2}\right| \geq f(n)$。