Putnam 1980 A4

a) Prove that there exist integers $a, b, c$ not all zero and each of absolute value less than one million, such that $$|a +b \sqrt{2} +c \sqrt{3} | <10^{-11} .$$ b) Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that $$|a +b \sqrt{2} +c \sqrt{3} | >10^{-21} .$$