Problem

Source: Romanian National Olympiad 2015, grade xii, p. 2

Tags: Ring Theory, Field theory, superior algebra



Show that the set of all elements minus $ 0 $ of a finite division ring that has at least $ 4 $ elements can be partitioned into two nonempty sets $ A,B $ having the property that $$ \sum_{x\in A} x=\prod_{y\in B} y. $$