2004
Problem - 4328
Let $m$ and $n$ be positive integers. Show that $$\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$$
This inequality is equivalent to $$(m+n)^{m+n} > \frac{(m+n)!}{m!n!}m^mn^n =\binom{m+n}{m,n}m^mn^n$$
which indeed holds because the expansion of the left side contains the right side plus some other positive terms.