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Contests
National and Regional Contests
Bosnia Herzegovina Contests
Regional Olympiad - Republic of Srpska
2004 Regional Olympiad - Republic of Srpska
2
interesting one
interesting one
Source: RS2004
March 20, 2005
inequalities proposed
inequalities
Problem Statement
The positive real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfy
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
. Show that
3
x
y
z
(
1
x
+
1
y
+
1
z
+
1
1
−
x
+
1
1
−
y
+
1
1
−
z
)
≥
4
+
4
x
y
z
(
1
−
x
)
(
1
−
y
)
(
1
−
z
)
.
\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+ \frac{4xyz}{(1-x)(1-y)(1-z)}.
3
x
yz
(
x
1
+
y
1
+
z
1
+
1
−
x
1
+
1
−
y
1
+
1
−
z
1
)
≥
4
+
(
1
−
x
)
(
1
−
y
)
(
1
−
z
)
4
x
yz
.
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