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Polish MO Finals
2012 Polish MO Finals
6
Inequality with squares
Inequality with squares
Source: 63 Polish MO 2012 Finals - Problem 6
April 23, 2018
algebra
Inequality
Poland
inequalities
Problem Statement
Show that for any positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
true is inequality:
(
a
−
b
c
)
2
+
(
b
−
c
a
)
2
+
(
c
−
a
b
)
2
≥
2
2
(
a
−
b
c
+
b
−
c
a
+
c
−
a
b
)
\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)
(
c
a
−
b
)
2
+
(
a
b
−
c
)
2
+
(
b
c
−
a
)
2
≥
2
2
(
c
a
−
b
+
a
b
−
c
+
b
c
−
a
)
.
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