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2019 JBMO Shortlist
C5
C5
Part of
2019 JBMO Shortlist
Problems
(1)
JBMO Shortlist 2019 C5
Source:
9/12/2020
An economist and a statistician play a game on a calculator which does only one operation. The calculator displays only positive integers and it is used in the following way: Denote by
n
n
n
an integer that is shown on the calculator. A person types an integer,
m
m
m
, chosen from the set
{
1
,
2
,
.
.
.
,
99
}
\{ 1, 2, . . . , 99 \}
{
1
,
2
,
...
,
99
}
of the first
99
99
99
positive integers, and if
m
%
m\%
m
%
of the number
n
n
n
is again a positive integer, then the calculator displays
m
%
m\%
m
%
of
n
n
n
. Otherwise, the calculator shows an error message and this operation is not allowed. The game consists of doing alternatively these operations and the player that cannot do the operation looses. How many numbers from
{
1
,
2
,
.
.
.
,
2019
}
\{1, 2, . . . , 2019\}
{
1
,
2
,
...
,
2019
}
guarantee the winning strategy for the statistician, who plays second?For example, if the calculator displays
1200
1200
1200
, the economist can type
50
50
50
, giving the number
600
600
600
on the calculator, then the statistician can type
25
25
25
giving the number
150
150
150
. Now, for instance, the economist cannot type
75
75
75
as
75
%
75\%
75%
of
150
150
150
is not a positive integer, but can choose
40
40
40
and the game continues until one of them cannot type an allowed numberProposed by Serbia
combinatorics