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Ecuador Contests
Ecuador Mathematical Olympiad (OMEC)
2020 Ecuador NMO (OMEC)
6
6
Part of
2020 Ecuador NMO (OMEC)
Problems
(1)
Guayaco board combi nt problem
Source: OMEC Ecuador National Olympiad Final Round 2020 N3 P6 day 2
11/11/2024
A board
1
1
1
x
k
k
k
is called guayaco if: -Each unit square is painted with exactly one of
k
k
k
available colors. -If
g
c
d
(
i
,
k
)
>
1
gcd(i,k)>1
g
c
d
(
i
,
k
)
>
1
, the
i
i
i
th unit square is painted with the same color as
(
i
−
1
)
(i-1)
(
i
−
1
)
th unit square. -If
g
c
d
(
i
,
k
)
=
1
gcd(i, k)=1
g
c
d
(
i
,
k
)
=
1
, the
i
i
i
th unit square is painted with the same color as
(
k
−
i
)
(k-i)
(
k
−
i
)
th unit square. Sebastian chooses a positive integer
a
a
a
and calculates the number of boards
1
1
1
x
a
a
a
that are guayacos. After that, David chooses a positive integer
b
b
b
and calculates the number of boards
1
1
1
x
b
b
b
that are guayacos. David wins if the number of boards
1
1
1
x
a
a
a
that are guayacos is the same as the number of boards
1
1
1
x
b
b
b
that are guayacos, otherwise, Sebastian wins. Find all the pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
such that, with those numbers, David wins.
combinatorics