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Argentina Contests
Argentina National Olympiad
2019 Argentina National Olympiad
4
4
Part of
2019 Argentina National Olympiad
Problems
(1)
a\sqrt2 is rational if a^2+b \sqrt2 is a rational number
Source: 2019 Argentina OMA Finals L3 p4
1/16/2023
If we have a set
M
M
M
of
2019
2019
2019
real numbers such that for every even
a
a
a
,
b
b
b
of numbers of
M
M
M
it is verified that
a
2
+
b
2
a^2+b \sqrt2
a
2
+
b
2
is a rational number. Show that for all
a
a
a
of
M
M
M
,
a
2
a\sqrt2
a
2
is a rational number.
rational
algebra