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International Contests
Balkan MO Shortlist
2014 Balkan MO Shortlist
C2
C2
Part of
2014 Balkan MO Shortlist
Problems
(1)
Nonempty subsets and circle !
Source:
5/16/2015
Let
M
=
{
1
,
2
,
.
.
.
,
2013
}
M=\{1,2,...,2013\}
M
=
{
1
,
2
,
...
,
2013
}
and let
Γ
\Gamma
Γ
be a circle. For every nonempty subset
B
B
B
of the set
M
M
M
, denote by
S
(
B
)
S(B)
S
(
B
)
sum of elements of the set
B
B
B
, and define
S
(
∅
)
=
0
S(\varnothing)=0
S
(
∅
)
=
0
(
∅
\varnothing
∅
is the empty set ). Is it possible to join every subset
B
B
B
of
M
M
M
with some point
A
A
A
on the circle
Γ
\Gamma
Γ
so that following conditions are fulfilled:
1
1
1
. Different subsets are joined with different points;
2
2
2
. All joined points are vertices of a regular polygon;
3
3
3
. If
A
1
,
A
2
,
.
.
.
,
A
k
A_1,A_2,...,A_k
A
1
,
A
2
,
...
,
A
k
are some of the joined points,
k
>
2
k>2
k
>
2
, such that
A
1
A
2
.
.
.
A
k
A_1A_2...A_k
A
1
A
2
...
A
k
is a regular
k
−
g
o
n
k-gon
k
−
g
o
n
, then
2014
2014
2014
divides
S
(
B
1
)
+
S
(
B
2
)
+
.
.
.
+
S
(
B
k
)
S(B_1)+S(B_2)+...+S(B_k)
S
(
B
1
)
+
S
(
B
2
)
+
...
+
S
(
B
k
)
?
combinatorics