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National and Regional Contests
Azerbaijan Contests
Azerbaijan Team Selection Test
2018 Azerbaijan IMO TST
2018 Azerbaijan IMO TST
Part of
Azerbaijan Team Selection Test
Subcontests
(1)
1
1
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Putting two colored papers into folders
Let
m
m
m
and
n
n
n
be natural numbers. Professor Mubariz has
m
m
m
folders and Professor Nazim has
n
n
n
folders; initially, all folders are empty. Every day, where the day numbers are marked as
d
=
1
,
2
,
3....
,
d = 1,2,3 ....,
d
=
1
,
2
,
3....
,
Prof. Mubariz is given
2018
2018
2018
blue papers, and Prof. Nazim is given
2018
2018
2018
orange papers. On day
d
(
d
=
1
,
2
,
3
,
.
.
.
)
,
d ( d = 1, 2, 3, ...),
d
(
d
=
1
,
2
,
3
,
...
)
,
they both perform the following operations:[*] If the
2018
2018
2018
papers given to this professor are not enough to place
d
d
d
papers in each of his folders, then he distributes all the
2018
2018
2018
papers given to him to his students. If the
2018
2018
2018
papers given to this professor are enough to place
d
d
d
papers in each of his folders, firstly, he places
d
d
d
papers in each of his folders. [*] If this professor still has papers left after the first step, he places them in the other professor's folders, with the same number in each folder and as many as possible. [*] If this professor still has papers left after the second step, he distributes them to his students.Prove that after
6
6
6
years, the number of blue papers in one folder of Prof. Nazim will be equal to the number of orange papers in one folder of Prof. Mubariz.