MathDB
Problems
Contests
International Contests
Balkan MO Shortlist
2021 Balkan MO Shortlist
N1
BMO Shortlist 2021 N1
BMO Shortlist 2021 N1
Source: BMO Shortlist 2021
May 8, 2022
Balkan
shortlist
2021
number theory
arithmetic mean
Problem Statement
Let
n
≥
2
n \geq 2
n
≥
2
be an integer and let
M
=
{
a
1
+
a
2
+
.
.
.
+
a
k
k
:
1
≤
k
≤
n
and
1
≤
a
1
<
…
<
a
k
≤
n
}
M=\bigg\{\frac{a_1 + a_2 + ... + a_k}{k}: 1 \le k \le n\text{ and }1 \le a_1 < \ldots < a_k \le n\bigg\}
M
=
{
k
a
1
+
a
2
+
...
+
a
k
:
1
≤
k
≤
n
and
1
≤
a
1
<
…
<
a
k
≤
n
}
be the set of the arithmetic means of the elements of all non-empty subsets of
{
1
,
2
,
.
.
.
,
n
}
\{1, 2, ..., n\}
{
1
,
2
,
...
,
n
}
. Find
min
{
∣
a
−
b
∣
:
a
,
b
∈
M
with
a
≠
b
}
.
\min\{|a - b| : a, b \in M\text{ with } a \neq b\}.
min
{
∣
a
−
b
∣
:
a
,
b
∈
M
with
a
=
b
}
.
Back to Problems
View on AoPS