China Mathematical Competition

Obviously, $$f(n)=\frac{1}{n+1}+\frac{1}{n+2}+\cdots +\frac{1}{2n+1}$$ is monotonically decreasing. Therefore, $f(1)$ reaches the maximum of $f(n)$. From $$f(1)=\frac{1}{2}+\frac{1}{3}<a-2007\frac{1}{3},$$ we have $a>2008$. Therefore, the least positive integer of $a$ is $2009$.