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2022 JBMO Shortlist
A6
JBMO Shortlist 2022 A6
JBMO Shortlist 2022 A6
Source: JBMO Shortlist 2022
June 26, 2023
Inequality
Junior
Balkan
shortlist
algebra
absolute value
Problem Statement
Let
a
,
b
,
a, b,
a
,
b
,
and
c
c
c
be positive real numbers such that
a
2
+
b
2
+
c
2
=
3
a^2 + b^2 + c^2 = 3
a
2
+
b
2
+
c
2
=
3
. Prove that
a
2
+
b
2
2
a
b
+
b
2
+
c
2
2
b
c
+
c
2
+
a
2
2
c
a
+
2
(
a
b
+
b
c
+
c
a
)
3
≥
5
+
∣
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
∣
.
\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 + |(a - b)(b - c)(c - a)|.
2
ab
a
2
+
b
2
+
2
b
c
b
2
+
c
2
+
2
c
a
c
2
+
a
2
+
3
2
(
ab
+
b
c
+
c
a
)
≥
5
+
∣
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
∣.
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