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Contests
National and Regional Contests
Canada Contests
Canadian Open Math Challenge
2024 Canadian Open Math Challenge
B4
B4
Part of
2024 Canadian Open Math Challenge
Problems
(1)
2024 COMC B4
Source:
11/4/2024
Initially, the integer
80
80
80
is written on a blackboard. At each step, the integer
x
x
x
on the blackboard is replaced with an integer chosen uniformly at random among [0,x−1], unless
x
=
0
x=0
x
=
0
, in which case it is replaced by an integer chosen uniformly at random among [0,2024]. Let
P
(
a
,
b
)
P(a,b)
P
(
a
,
b
)
be the probability that after
a
a
a
steps, the integer on the board is
b
b
b
. Determine
lim
x
→
∞
P
(
a
,
80
)
P
(
a
,
2024
)
\lim_{x\to\infty}\frac{P(a,80)}{P(a,2024)}
x
→
∞
lim
P
(
a
,
2024
)
P
(
a
,
80
)
(that is, the value that the function
P
(
a
,
80
)
P
(
a
,
2024
)
\frac{P(a,80)}{P(a,2024)}
P
(
a
,
2024
)
P
(
a
,
80
)
approaches as
a
a
a
goes to infinity).
Comc