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National and Regional Contests
Greece Contests
Greece JBMO TST
2009 Greece JBMO TST
3
3
Part of
2009 Greece JBMO TST
Problems
(1)
(a\sqrt2+b\sqrt3)/(b\sqrt2+c\sqrt3) \in Q => (a^2+b^2+c^2)/(a+b+c) in Z
Source: Greece JBMO TST 2009 p3
4/29/2019
Given are the non zero natural numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that the number
a
2
+
b
3
b
2
+
c
3
\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}
b
2
+
c
3
a
2
+
b
3
is rational. Prove that the number
a
2
+
b
2
+
c
2
a
+
b
+
c
\frac{a^2+b^2+c^2}{a+b+c}
a
+
b
+
c
a
2
+
b
2
+
c
2
is an integer .
algebra
rational
integration
radical