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Balkan MO Shortlist
2022 Balkan MO Shortlist
A3
Inequality with strange condition
Inequality with strange condition
Source: BMO Shortlist 2022, A3
May 13, 2023
algebra
inequalities
Problem Statement
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be non-negative real numbers such that
1
a
+
1
+
1
b
+
1
+
1
c
+
1
+
1
d
+
1
=
3.
\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=3.
a
+
1
1
+
b
+
1
1
+
c
+
1
1
+
d
+
1
1
=
3.
Prove that
3
(
a
b
+
b
c
+
c
a
+
a
d
+
b
d
+
c
d
)
+
4
a
+
b
+
c
+
d
⩽
5.
3(ab+bc+ca+ad+bd+cd)+\frac{4}{a+b+c+d}\leqslant 5.
3
(
ab
+
b
c
+
c
a
+
a
d
+
b
d
+
c
d
)
+
a
+
b
+
c
+
d
4
⩽
5.
Vasile Cîrtoaje and Leonard Giugiuc
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