Putnam 1980, problem A4:
Each of $a, b, c$ are integers and none have an absolute value greater than or equal to one million. Prove that $$ \left| a + b\sqrt2 + c\sqrt3\right| > 10^{-21} $$ for any choice of $a, b, c$.
I would have never thought this would have anything to do with combinatorics or the pigeonhole principle but in the book 'Problem solving strategies' by Arthur Engel it's been included in the chapter on the pigeonhole principle. I would give an attempt but I have no idea. I thought about looking at the fractional parts of each of the three terms, but that didn't lead to anything. If you have a solution please give hints and not the whole solution. Thank you