Question 1
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 $|a_i-1/2|\ge f(n)$.