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National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2010 Vietnam Team Selection Test
1
VN TST 2010 Pro 4
VN TST 2010 Pro 4
Source:
October 24, 2010
inequalities
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive integers which satisfy the condition:
16
(
a
+
b
+
c
)
≥
1
a
+
1
b
+
1
c
16(a+b+c)\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}
16
(
a
+
b
+
c
)
≥
a
1
+
b
1
+
c
1
. Prove that
∑
c
y
c
(
1
a
+
b
+
2
a
+
2
c
)
3
≤
8
9
\sum_{cyc} \left( \frac{1}{a+b+\sqrt{2a+2c}} \right)^{3}\leq \frac{8}{9}
cyc
∑
(
a
+
b
+
2
a
+
2
c
1
)
3
≤
9
8
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