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Romanian Masters of Mathematics Collection
2019 Romanian Master of Mathematics Shortlist
A2
A2
Part of
2019 Romanian Master of Mathematics Shortlist
Problems
(1)
Inequality with permutations
Source: 2019 Belarus Team Selection Test 7.3
9/2/2019
Given a positive integer
n
n
n
, determine the maximal constant
C
n
C_n
C
n
satisfying the following condition: for any partition of the set
{
1
,
2
,
…
,
2
n
}
\{1,2,\ldots,2n \}
{
1
,
2
,
…
,
2
n
}
into two
n
n
n
-element subsets
A
A
A
and
B
B
B
, there exist labellings
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
and
b
1
,
b
2
,
…
,
b
n
b_1,b_2,\ldots,b_n
b
1
,
b
2
,
…
,
b
n
of
A
A
A
and
B
B
B
, respectively, such that
(
a
1
−
b
1
)
2
+
(
a
2
−
b
2
)
2
+
…
+
(
a
n
−
b
n
)
2
≥
C
n
.
(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
(
a
1
−
b
1
)
2
+
(
a
2
−
b
2
)
2
+
…
+
(
a
n
−
b
n
)
2
≥
C
n
.
(B. Serankou, M. Karpuk)
inequalities
algebra