MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2018 Moldova Team Selection Test
10
10
Part of
2018 Moldova Team Selection Test
Problems
(1)
Intersting inequality
Source: Moldova TST 2018
4/6/2018
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
.
inequalities