MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2009 Turkey MO (2nd round)
2
Turkey NMO 2009 Q5
Turkey NMO 2009 Q5
Source:
August 31, 2010
inequalities
calculus
derivative
function
inequalities proposed
Problem Statement
Show that
(
b
+
c
)
(
a
4
−
b
2
c
2
)
a
b
+
2
b
c
+
c
a
+
(
c
+
a
)
(
b
4
−
c
2
a
2
)
b
c
+
2
c
a
+
a
b
+
(
a
+
b
)
(
c
4
−
a
2
b
2
)
c
a
+
2
a
b
+
b
c
≥
0
\frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0
ab
+
2
b
c
+
c
a
(
b
+
c
)
(
a
4
−
b
2
c
2
)
+
b
c
+
2
c
a
+
ab
(
c
+
a
)
(
b
4
−
c
2
a
2
)
+
c
a
+
2
ab
+
b
c
(
a
+
b
)
(
c
4
−
a
2
b
2
)
≥
0
for all positive real numbers
a
,
b
,
c
.
a, \: b , \: c.
a
,
b
,
c
.
Back to Problems
View on AoPS