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National and Regional Contests
Greece Contests
Greece National Olympiad
2007 Greece National Olympiad
2
2
Part of
2007 Greece National Olympiad
Problems
(1)
messy ineq with triangle
Source: Greek MO 2007
3/5/2007
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be sides of a triangle, show that
(
c
+
a
−
b
)
4
a
(
a
+
b
−
c
)
+
(
a
+
b
−
c
)
4
b
(
b
+
c
−
a
)
+
(
b
+
c
−
a
)
4
c
(
c
+
a
−
b
)
≥
a
b
+
b
c
+
c
a
.
\frac{(c+a-b)^{4}}{a(a+b-c)}+\frac{(a+b-c)^{4}}{b(b+c-a)}+\frac{(b+c-a)^{4}}{c(c+a-b)}\geq ab+bc+ca.
a
(
a
+
b
−
c
)
(
c
+
a
−
b
)
4
+
b
(
b
+
c
−
a
)
(
a
+
b
−
c
)
4
+
c
(
c
+
a
−
b
)
(
b
+
c
−
a
)
4
≥
ab
+
b
c
+
c
a
.
inequalities
inequalities unsolved