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Moldova Team Selection Test
2013 Moldova Team Selection Test
4
Nine number inequality
Nine number inequality
Source: Moldova TST 2013
April 16, 2013
inequalities
inequalities unsolved
Problem Statement
Prove that for any positive real numbers
a
i
,
b
i
,
c
i
a_i,b_i,c_i
a
i
,
b
i
,
c
i
with
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
,
(
a
1
3
+
b
1
3
+
c
1
3
+
1
)
(
a
2
3
+
b
2
3
+
c
2
3
+
1
)
(
a
3
3
+
b
3
3
+
c
3
3
+
1
)
≥
3
4
(
a
1
+
b
1
+
c
1
)
(
a
2
+
b
2
+
c
2
)
(
a
3
+
b
3
+
c
3
)
(a_1^3+b_1^3+c_1^3+1)(a_2^3+b_2^3+c_2^3+1)(a_3^3+b_3^3+c_3^3+1)\geq \frac{3}{4} (a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)
(
a
1
3
+
b
1
3
+
c
1
3
+
1
)
(
a
2
3
+
b
2
3
+
c
2
3
+
1
)
(
a
3
3
+
b
3
3
+
c
3
3
+
1
)
≥
4
3
(
a
1
+
b
1
+
c
1
)
(
a
2
+
b
2
+
c
2
)
(
a
3
+
b
3
+
c
3
)
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