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Difference between revisions of "2013 AMC 12A Problems/Problem 21"

(Solution 2 is equivalent to the original solution, and assumes the answer a priori)
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<math>(\log 2016, \log 2017)</math>
 
<math>(\log 2016, \log 2017)</math>
 
==Solution 2==
 
 
We see that <math>10^A = 2013 + \cdots</math>, and that the quantity within "<math>\cdots</math>" is between <math>\log(1000)</math> and <math>\log(10000)</math>, i.e. <math>(3,4)</math>, therefore <math>10^A \in (2016,2017)</math>, or <math>A\in (\log(2016),\log(2017))</math>.
 

Revision as of 21:58, 13 February 2013

Solution

Let $f(x) = \log(x + f(x-1))$ and $f(2) = log(2)$, and from the problem description, $A = f(2013)$

We can reason out an approximation, by ignoring the $f(x-1)$:

$f(x) \approx \log x$

And a better approximation, by plugging in our first approximation for $f(x-1)$ in our original definition for $f(x)$:

$f(x) \approx \log(x + \log(x-1))$

And an even better approximation:

$f(x) \approx \log(x + \log(x-1 + \log(x-2)))$

Continuing this pattern, obviously, will eventually terminate at our original definition of $f(x)$.

However, at $x = 2013$, going further than our second approximation will not distinguish between our answer choices.

So we take our second approximation and plug in.

$f(2013) \approx \log(2013 + \log 2012)$

Since $1000 < 2012 < 10000$, we know $3 < log(2012) < 4$. This gives us our answer range:

$\log 2016 < \log(2013 + \log 2012) < \log(2017)$

$(\log 2016, \log 2017)$

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