Problem

Source: Iran 3rd round 2011-Number Theory exam-P2

Tags: floor function, number theory proposed, number theory



Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition: \[|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.\] Proposed by Mohammad Gharakhani