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\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2}, diophantine inequality

Source: 2003 Romania JBMO TST 1.3

June 1, 2020

diophantinenumber theoryradicalinequalities

Problem Statement

Let

n

be a positive integer. Prove that there are no positive integers

x

and

y

such as

\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2}