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2012 USAJMO
3
USAJMO problem 3: Inequality
USAJMO problem 3: Inequality
Source:
April 24, 2012
AMC
USA(J)MO
USAJMO
inequalities
algebra
polynomial
ratio
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers. Prove that
a
3
+
3
b
3
5
a
+
b
+
b
3
+
3
c
3
5
b
+
c
+
c
3
+
3
a
3
5
c
+
a
≥
2
3
(
a
2
+
b
2
+
c
2
)
\frac{a^3+3b^3}{5a+b}+\frac{b^3+3c^3}{5b+c}+\frac{c^3+3a^3}{5c+a} \geq \frac{2}{3}(a^2+b^2+c^2)
5
a
+
b
a
3
+
3
b
3
+
5
b
+
c
b
3
+
3
c
3
+
5
c
+
a
c
3
+
3
a
3
≥
3
2
(
a
2
+
b
2
+
c
2
)
.
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