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Greece National Olympiad
1998 Greece National Olympiad
3
a,b,c non-zero real numbers
a,b,c non-zero real numbers
Source:
December 17, 2009
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inequalities unsolved
inequalities
Problem Statement
Prove that for any non-zero real numbers
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
(
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 35.
(
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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