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PEN F Problems
6
F 6
F 6
Source:
May 25, 2007
rational numbers
Problem Statement
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
non-zero real numbers such that
x
y
xy
x
y
,
y
z
yz
yz
,
z
x
zx
z
x
are rational. [*] Show that the number
x
2
+
y
2
+
z
2
x^{2}+y^{2}+z^{2}
x
2
+
y
2
+
z
2
is rational. [*] If the number
x
3
+
y
3
+
z
3
x^{3}+y^{3}+z^{3}
x
3
+
y
3
+
z
3
is also rational, show that
x
x
x
,
y
y
y
,
z
z
z
are rational.
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