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Balkan MO Shortlist
2019 Balkan MO Shortlist
A5
(ab)^2 + (bc)^2 + (ca)^2
(ab)^2 + (bc)^2 + (ca)^2
Source: Shortlist BMO 2019, A5
November 7, 2020
Inequality
algebra
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers, such that
(
a
b
)
2
+
(
b
c
)
2
+
(
c
a
)
2
=
3
(ab)^2 + (bc)^2 + (ca)^2 = 3
(
ab
)
2
+
(
b
c
)
2
+
(
c
a
)
2
=
3
. Prove that
(
a
2
−
a
+
1
)
(
b
2
−
b
+
1
)
(
c
2
−
c
+
1
)
≥
1.
(a^2 - a + 1)(b^2 - b + 1)(c^2 - c + 1) \ge 1.
(
a
2
−
a
+
1
)
(
b
2
−
b
+
1
)
(
c
2
−
c
+
1
)
≥
1.
Proposed by Florin Stanescu (wer), România
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