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National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2019 Junior Balkan Team Selection Tests - Moldova
8
8
Part of
2019 Junior Balkan Team Selection Tests - Moldova
Problems
(1)
Moldova JTST 2019 P8
Source:
4/19/2019
It is considered a regular polygon with
n
n
n
sides, where
n
(
n
>
3
)
n(n>3)
n
(
n
>
3
)
is an odd number that does not divide by 3. From the vertices of the polygon are arbitrarily chosen
m
(
0
≤
m
≤
n
)
m(0\leq m\leq n)
m
(
0
≤
m
≤
n
)
vertices that are colored in red and the others in black. A triangle with the vertices at the vertices of the polygon it is considered
m
o
n
o
c
o
l
o
r
monocolor
m
o
n
oco
l
or
,if all of its vertices are of the same color. Prove that the number of all
m
o
n
o
c
o
l
o
r
monocolor
m
o
n
oco
l
or
isosceles triangles with the vertices at the given polygon ends does not depend on the way of coloring of the vertices of the polygon. Determine the number of all these
m
o
n
o
c
o
l
o
r
monocolor
m
o
n
oco
l
or
isosceles triangles.
combinatorics