Difference between revisions of "2020 CIME II Problems/Problem 4"
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==Similar Problems== | ==Similar Problems== | ||
[[2009 AIME I Problems/Problem 6|2009 AIME I Problem 6]] | [[2009 AIME I Problems/Problem 6|2009 AIME I Problem 6]] | ||
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| + | The intuition behind the above solution: | ||
| + | consider 7^3 = 343 <= N < 8^3 = 512 ; notice that any N in this range will have 4 digits in base 7 ( since 1*7^3 <= N) and since 1*8^3 > N , we have almost 3 digits in base 8 representation of N | ||
| + | |||
| + | its easy to see 3 can be replaced by any positive integer k , where k = 1,2,3 | ||
==See also== | ==See also== | ||
Latest revision as of 23:49, 18 December 2020
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 ![\[\sum^{3}_{k=1}[8^k-7^k]=8-7+64-49+512-343=185.\]](http://latex.artofproblemsolving.com/1/7/3/1739855fc74562170b4bb3610ab65377fbc07200.png)
The requested probability is ![\[\frac{185}{999}=\frac{5}{27},\]](http://latex.artofproblemsolving.com/4/6/2/462894a34dabff95b02534749bf0a5e7dcdd68bf.png)
and so the answer is 
.
Similar Problems
The intuition behind the above solution:
consider 7^3 = 343 <= N < 8^3 = 512 ; notice that any N in this range will have 4 digits in base 7 ( since 1*7^3 <= N) and since 1*8^3 > N , we have almost 3 digits in base 8 representation of N
its easy to see 3 can be replaced by any positive integer k , where k = 1,2,3
See also
| 2020 CIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All CIME Problems and Solutions | ||
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