Asia Pacific Mathematics Olympiad (APMO) competition problems

The answer is
$$f(n)= \begin{cases}0 & \text{ if } n \text{ is even } \\ \frac{1}{2 n} & \text{ if } n \text{ is odd }\end{cases}$$
First, assume that $n$ is even. If $a_{i}=\frac{1}{2}$ for all $i$, then the sum $a_{1}+a_{2}+\cdots+a_{n}$ is an integer. Since $\left|a_{i}-\frac{1}{2}\right|=0$ for all $i$, we may conclude $f(n)=0$ for any even $n$.

Now assume that $n$ is odd. Suppose that $\left|a_{i}-\frac{1}{2}\right|<\frac{1}{2 n}$ for all $1 \leq i \leq n$. Then, since $\sum_{i=1}^{n} a_{i}$ is an integer,
$$\frac{1}{2} \leq\left|\sum_{i=1}^{n} a_{i}-\frac{n}{2}\right| \leq \sum_{i=1}^{n}\left|a_{i}-\frac{1}{2}\right|<\frac{1}{2 n} \cdot n=\frac{1}{2}$$
a contradiction. Thus $\left|a_{i}-\frac{1}{2}\right| \geq \frac{1}{2 n}$ for some $i$, as required. On the other hand, putting $n=2 m+1$ and $a_{i}=\frac{m}{2 m+1}$ for all $i$ gives $\sum a_{i}=m$, while
$$\left|a_{i}-\frac{1}{2}\right|=\frac{1}{2}-\frac{m}{2 m+1}=\frac{1}{2(2 m+1)}=\frac{1}{2 n}$$
for all $i$. Therefore, $f(n)=\frac{1}{2 n}$ is the best possible for any odd $n$.