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International Contests
Kvant Problems
Kvant 2020
M942
Two disjoints sets
Two disjoints sets
Source:
September 20, 2021
Sum
Problem Statement
We divide the set
{
1
,
2
,
⋯
,
2
n
}
\{1,2,\cdots,2n\}
{
1
,
2
,
⋯
,
2
n
}
into two disjoint sets :
{
a
1
,
a
2
,
⋯
,
a
n
}
\{a_1,a_2,\cdots,a_n\}
{
a
1
,
a
2
,
⋯
,
a
n
}
and
{
b
1
,
b
2
,
⋯
,
b
n
}
\{b_1,b_2,\cdots,b_n\}
{
b
1
,
b
2
,
⋯
,
b
n
}
such that :
a
1
<
a
2
<
⋯
<
a
n
and
b
1
>
b
2
>
⋯
>
b
n
.
a_1<a_2<\cdots<a_n\text{ and } b_1>b_2>\cdots>b_n.
a
1
<
a
2
<
⋯
<
a
n
and
b
1
>
b
2
>
⋯
>
b
n
.
Show that :
∣
a
1
−
b
1
∣
+
⋯
+
∣
a
n
−
b
n
∣
=
n
2
.
|a_1-b_1|+\cdots+|a_n-b_n|=n^2.
∣
a
1
−
b
1
∣
+
⋯
+
∣
a
n
−
b
n
∣
=
n
2
.
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