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2004 Moldova Team Selection Test
9
2004 Republic of Moldova , Problem 9
2004 Republic of Moldova , Problem 9
Source: Crux 2007
February 2, 2015
inequalities
inequalities proposed
Problem Statement
Let
a
,
b
a,b
a
,
b
and
c
c
c
be positive real numbers . Prove that
∣
4
(
b
3
−
c
3
)
b
+
c
+
4
(
c
3
−
a
3
)
c
+
a
+
4
(
a
3
−
b
3
)
a
+
b
∣
≤
(
b
−
c
)
2
+
(
c
−
a
)
2
+
(
a
−
b
)
2
.
\left | \frac{4(b^3-c^3)}{b+c}+ \frac{4(c^3-a^3)}{c+a}+ \frac{4(a^3-b^3)}{a+b} \right |\leq (b-c)^2+(c-a)^2+(a-b)^2.
b
+
c
4
(
b
3
−
c
3
)
+
c
+
a
4
(
c
3
−
a
3
)
+
a
+
b
4
(
a
3
−
b
3
)
≤
(
b
−
c
)
2
+
(
c
−
a
)
2
+
(
a
−
b
)
2
.
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