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National and Regional Contests
China Contests
China Northern MO
2006 China Northern MO
5
a+b+c = 3
a+b+c = 3
Source: China North MO
August 14, 2006
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inequalities
inequalities unsolved
China
Problem Statement
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive numbers such that
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
, show that:
a
2
+
9
2
a
2
+
(
b
+
c
)
2
+
b
2
+
9
2
b
2
+
(
a
+
c
)
2
+
c
2
+
9
2
c
2
+
(
a
+
b
)
2
≤
5
\frac{a^{2}+9}{2a^{2}+(b+c)^{2}}+\frac{b^{2}+9}{2b^{2}+(a+c)^{2}}+\frac{c^{2}+9}{2c^{2}+(a+b)^{2}}\leq 5
2
a
2
+
(
b
+
c
)
2
a
2
+
9
+
2
b
2
+
(
a
+
c
)
2
b
2
+
9
+
2
c
2
+
(
a
+
b
)
2
c
2
+
9
≤
5
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