TITLE: Proving that $\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\frac{1}{2}n^n$
QUESTION [18 upvotes]: How can we prove that
$$\displaystyle\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\dfrac{1}{2}n^n$$
where $\displaystyle\binom{n}{i}=\dfrac{n!}{i!(n-i)!}$.
This inequality is very interesting. I think this problem has nice methods, but my method is very ugly.

REPLY [2 votes]: This was Question 10 from the 2009 Sydney University Mathematics Society (SUMS) problem solving competition, see here. A combinatorial solution interpreting the problem in terms of trees can be found in the last section of the solutions here.