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National and Regional Contests
Argentina Contests
Argentina National Olympiad
2023 Argentina National Olympiad
5
5
Part of
2023 Argentina National Olympiad
Problems
(1)
ARGENTINA MO 2023 National Level 3
Source:
3/22/2024
Let
n
n
n
be a positive integer. Beto writes a list of
n
n
n
non-negative integers on the board. Then he performs a succession of moves (two steps) of the following type: First for each
i
=
1
,
2
,
.
.
.
,
n
i=1,2,...,n
i
=
1
,
2
,
...
,
n
, he counts how many numbers on the board are less than or equal to
i
i
i
. Let
a
i
a_i
a
i
be the number obtained for each
i
=
1
,
2
,
.
.
.
,
n
i=1,2,...,n
i
=
1
,
2
,
...
,
n
. Next, he erases all the numbers from the board and writes the numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
. For example, if
n
=
5
n=5
n
=
5
and the initial numbers on the board are
0
,
7
,
2
,
6
,
2
0,7,2,6,2
0
,
7
,
2
,
6
,
2
, after the first move, the numbers on the board will bec
1
,
3
,
3
,
3
,
3
1,3,3,3,3
1
,
3
,
3
,
3
,
3
;after the second move they will be
1
,
1
,
5
,
5
,
5
1,1,5,5,5
1
,
1
,
5
,
5
,
5
, and so on.
a
)
a)
a
)
Show that, for every
n
n
n
and every initial configuration, there will come a time after which the numbers will no longer be modified when using this move.
b
)
b)
b
)
Find (as a function of
n
n
n
) the minimum value of
k
k
k
such that, for any initial configuration, the moves made from move number
k
k
k
will not change the numbers on the board.
number theory