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a\sqrt2 is rational if a^2+b \sqrt2 is a rational number

Source: 2019 Argentina OMA Finals L3 p4

January 16, 2023
rationalalgebra

Problem Statement

If we have a set MM of 20192019 real numbers such that for every even aa, bb of numbers of MM it is verified that a2+b2a^2+b \sqrt2 is a rational number. Show that for all aa of MM, a2a\sqrt2 is a rational number.
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