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Danube Competition in Mathematics
2008 Danube Mathematical Competition
1
Danube Cup 2008
Danube Cup 2008
Source:
October 12, 2015
inequalities
Problem Statement
x
,
y
,
z
,
t
∈
R
+
∗
x,y,z,t \in \mathbb R_+^*
x
,
y
,
z
,
t
∈
R
+
∗
:
(
x
y
)
1
/
2
+
(
y
z
)
1
/
2
+
(
z
t
)
1
/
2
+
(
t
x
)
1
/
2
+
(
x
z
)
1
/
2
+
(
y
t
)
1
/
2
≥
3
(
x
y
z
+
x
y
t
+
x
z
t
+
y
z
t
)
1
3
(xy)^{1/2}+(yz)^{1/2}+(zt)^{1/2}+(tx)^{1/2}+(xz)^{1/2}+(yt)^{1/2} \ge 3(xyz+xyt+xzt+yzt)^{\frac{1}{3}}
(
x
y
)
1/2
+
(
yz
)
1/2
+
(
z
t
)
1/2
+
(
t
x
)
1/2
+
(
x
z
)
1/2
+
(
y
t
)
1/2
≥
3
(
x
yz
+
x
y
t
+
x
z
t
+
yz
t
)
3
1
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