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International Contests
Austrian-Polish
1991 Austrian-Polish Competition
5
5
Part of
1991 Austrian-Polish Competition
Problems
(1)
x^2+y^2+z^2 + xy+yz + zx> =2(\sqrt{x} +\sqrt{y}+ \sqrt{z}) if xyz = 1, x,y,z>0
Source: Austrian Polish 1991 APMC
5/1/2020
If
x
,
y
,
z
x,y, z
x
,
y
,
z
are arbitrary positive numbers with
x
y
z
=
1
xyz = 1
x
yz
=
1
, prove the inequality
x
2
+
y
2
+
z
2
+
x
y
+
y
z
+
z
x
≥
2
(
x
+
y
+
z
)
x^2+y^2+z^2 + xy+yz + zx \ge 2(\sqrt{x} +\sqrt{y}+ \sqrt{z})
x
2
+
y
2
+
z
2
+
x
y
+
yz
+
z
x
≥
2
(
x
+
y
+
z
)
.
inequalities
algebra