2020 CIME II Problems/Problem 4
Contents
Problem
The probability a randomly chosen positive integer has more digits when written in base
than when written in base
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Solution
If a positive integer has more digits in base
than base
, then
for some positive integer
. There are
positive integers
that satisfy this condition for every positive integer
. If
,
will be greater than
, so we only need to consider
. The number of possible values of
is
The requested probability is
and so the answer is
.
Similar Problems
See also
2020 CIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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