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2020 China Northern MO
BP1
Inequality on Positive Reals
Inequality on Positive Reals
Source: 2020 China North Mathematical Olympiad Basic Level P1
August 4, 2020
inequalities
algebra
Problem Statement
For all positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
, prove that
a
3
+
b
3
a
2
−
a
b
+
b
2
+
b
3
+
c
3
b
2
−
b
c
+
c
2
+
c
3
+
a
3
c
2
−
c
a
+
a
2
≥
2
(
a
2
+
b
2
+
c
2
)
\frac{a^3+b^3}{ \sqrt{a^2-ab+b^2} } + \frac{b^3+c^3}{ \sqrt{b^2-bc+c^2} } + \frac{c^3+a^3}{ \sqrt{c^2-ca+a^2} } \geq 2(a^2+b^2+c^2)
a
2
−
ab
+
b
2
a
3
+
b
3
+
b
2
−
b
c
+
c
2
b
3
+
c
3
+
c
2
−
c
a
+
a
2
c
3
+
a
3
≥
2
(
a
2
+
b
2
+
c
2
)
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