Difference between revisions of "2013 AIME II Problems/Problem 2"

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Note that <math>a</math> cannot be <math>0,</math> since that would cause the <math>\log_{2^a}</math> to have a <math>1</math> in the base, which is not possible.
 
Note that <math>a</math> cannot be <math>0,</math> since that would cause the <math>\log_{2^a}</math> to have a <math>1</math> in the base, which is not possible.
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==Video Solution==
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https://youtu.be/zf9ld5KL_g4
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~Lucas
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2013|n=II|num-b=1|num-a=3}}
 
{{AIME box|year=2013|n=II|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 17:21, 19 September 2022

Problem 2

Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000}))) = 0.\] Find the sum of all possible values of $a+b$.

Solution

To simplify, we write this logarithmic expression as an exponential one. Just looking at the first log, it has a base of 2 and an argument of the expression in parenthesis. Therefore, we can make 2 the base, 0 the exponent, and the argument the result. That means $\log_{2^a}(\log_{2^b}(2^{1000}))=1$ (because $2^0=1$). Doing this again, we get $\log_{2^b}(2^{1000})=2^a$. Doing the process one more time, we finally eliminate all of the logs, getting ${(2^{b})}^{(2^a)}=2^{1000}$. Using the property that ${(a^x)^{y}}=a^{xy}$, we simplify to $2^{b\cdot2^{a}}=2^{1000}$. Eliminating equal bases leaves $b\cdot2^a=1000$. The largest $a$ such that $2^a$ divides $1000$ is $3$, so we only need to check $1$,$2$, and $3$. When $a=1$, $b=500$; when $a=2$, $b=250$; when $a=3$, $b=125$. Summing all the $a$'s and $b$'s gives the answer of $\boxed{881}$.

Note that $a$ cannot be $0,$ since that would cause the $\log_{2^a}$ to have a $1$ in the base, which is not possible.

Video Solution

https://youtu.be/zf9ld5KL_g4 ~Lucas

See also

2013 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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