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Middle European Mathematical Olympiad
2023 Middle European Mathematical Olympiad
2
4-variable inequality
4-variable inequality
Source: MEMO 2023 T2
August 25, 2023
inequalities
algebra
Problem Statement
If
a
,
b
,
c
,
d
>
0
a, b, c, d>0
a
,
b
,
c
,
d
>
0
and
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
, show that
a
b
+
1
a
+
1
+
b
c
+
1
b
+
1
+
c
d
+
1
c
+
1
+
d
a
+
1
d
+
1
≥
4.
\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4.
a
+
1
ab
+
1
+
b
+
1
b
c
+
1
+
c
+
1
c
d
+
1
+
d
+
1
d
a
+
1
≥
4.
When does equality hold?
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