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National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2007 Hanoi Open Mathematics Competitions
8
HOMC (Vietnam)
HOMC (Vietnam)
Source:
January 28, 2016
inequalities
Problem Statement
Let a; b; c be positive integers. Prove that
(
b
+
c
−
a
)
2
(
b
+
c
)
2
+
a
2
+
(
c
+
a
−
b
)
2
(
c
+
a
)
2
+
b
2
+
(
a
+
b
−
c
)
2
(
a
+
b
)
2
+
c
2
≥
3
5
\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac{3}{5}
(
b
+
c
)
2
+
a
2
(
b
+
c
−
a
)
2
+
(
c
+
a
)
2
+
b
2
(
c
+
a
−
b
)
2
+
(
a
+
b
)
2
+
c
2
(
a
+
b
−
c
)
2
≥
5
3
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