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Infinitely many pairs of rational numbers

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

September 19, 2012
floor functionnumber theory proposednumber theory

Problem Statement

Prove that there exists infinitely many pairs of rational numbers (p1q,p2q)(\frac{p_1}{q},\frac{p_2}{q}) with p1,p2,q∈Np_1,p_2,q\in \mathbb N with the following condition: ∣3−p1q∣<q−32,∣2−p2q∣<q−32.|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.
Proposed by Mohammad Gharakhani
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