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Cauchy-Schwarz !
Cauchy-Schwarz !
Source:
February 4, 2015
Problem Statement
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers such that
a
+
b
+
c
=
a
2
+
b
2
+
c
2
a+b+c=a^2+b^2+c^2
a
+
b
+
c
=
a
2
+
b
2
+
c
2
. Prove that :
a
2
a
2
+
a
b
+
b
2
b
2
+
b
c
+
c
2
c
2
+
c
a
≥
a
+
b
+
c
2
\frac{a^2}{a^2+ab}+\frac{b^2}{b^2+bc}+\frac{c^2}{c^2+ca} \geq \frac{a+b+c}{2}
a
2
+
ab
a
2
+
b
2
+
b
c
b
2
+
c
2
+
c
a
c
2
≥
2
a
+
b
+
c
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