This week in the Logic Seminar in Keller 314, David Ross will give an easy proof of a slight extension of a result of Lagarias on the Diophantine equation
$$ c(1/x_1+cdots+1/x_s)+b/(x_1 x_2cdots x_s)=a$$
The proof will be nonstandard, but really only require a sufficiently-saturated ordered field extension of $mathbb R$.