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Balkan MO Shortlist
2023 Balkan MO Shortlist
A2
Inequality about existence of permutation
Inequality about existence of permutation
Source: BMO SL 2023 A2
May 3, 2024
algebra
Problem Statement
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be non-negative reals such that
1
a
+
3
+
1
b
+
3
+
1
c
+
3
+
1
d
+
3
=
1
\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}=1
a
+
3
1
+
b
+
3
1
+
c
+
3
1
+
d
+
3
1
=
1
. Show that there exists a permutation
(
x
1
,
x
2
,
x
3
,
x
4
)
(x_1, x_2, x_3, x_4)
(
x
1
,
x
2
,
x
3
,
x
4
)
of
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
, such that
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+
x
4
x
1
≥
4.
x_1x_2+x_2x_3+x_3x_4+x_4x_1 \geq 4.
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+
x
4
x
1
≥
4.
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