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Balkan MO
2019 Balkan MO
2
2
Part of
2019 Balkan MO
Problems
(1)
Inequality with condition a+b+c = ab+bc+ca (and special equality case)
Source: BMO 2019, problem 2
5/2/2019
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers such that
0
≤
a
≤
b
≤
c
0 \leq a \leq b \leq c
0
≤
a
≤
b
≤
c
and
a
+
b
+
c
=
a
b
+
b
c
+
c
a
>
0.
a+b+c=ab+bc+ca >0.
a
+
b
+
c
=
ab
+
b
c
+
c
a
>
0.
Prove that
b
c
(
a
+
1
)
≥
2
\sqrt{bc}(a+1) \geq 2
b
c
(
a
+
1
)
≥
2
and determine the equality cases.(Edit: Proposed by sir Leonard Giugiuc, Romania)
inequalities