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Moldova Team Selection Test
2018 Moldova Team Selection Test
10
Intersting inequality
Intersting inequality
Source: Moldova TST 2018
April 6, 2018
inequalities
Problem Statement
The positive real numbers
a
,
b
,
c
,
d
a,b, c,d
a
,
b
,
c
,
d
satisfy the equality
1
a
+
1
+
1
b
+
1
+
1
c
+
1
+
1
d
+
1
=
3
\frac {1}{a+1} + \frac {1}{b+1} + \frac {1}{c+1} + \frac{ 1}{d+1} = 3
a
+
1
1
+
b
+
1
1
+
c
+
1
1
+
d
+
1
1
=
3
. Prove the inequality
a
b
c
3
+
b
c
d
3
+
c
d
a
3
+
d
a
b
3
≤
4
3
\sqrt [3]{abc} + \sqrt [3]{bcd} + \sqrt [3]{cda} + \sqrt [3]{dab} \le \frac {4}{3}
3
ab
c
+
3
b
c
d
+
3
c
d
a
+
3
d
ab
≤
3
4
.
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