Difference between revisions of "2018 AMC 10A Problems/Problem 14"
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<cmath>2^{100} \leq 2^{96}*3^4.</cmath> | <cmath>2^{100} \leq 2^{96}*3^4.</cmath> | ||
This means that our original value, <math>x</math>, must be less than <math>81</math>. The only answer that is less than <math>81</math> is <math>80</math> so our answer is <math>\boxed{A}</math>. | This means that our original value, <math>x</math>, must be less than <math>81</math>. The only answer that is less than <math>81</math> is <math>80</math> so our answer is <math>\boxed{A}</math>. | ||
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Revision as of 14:44, 8 February 2018
What is the greatest integer less than or equal to ![\[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]](http://latex.artofproblemsolving.com/7/e/9/7e9fe91da45cd2ff16c0da3c9b0d7ba0a4f1d510.png)
Solution
Let's set this value equal to 
. We can write
![\[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}=x.\]](http://latex.artofproblemsolving.com/0/c/2/0c28b87f333620e9b3ee344cfcf661cc90e51901.png)
Multiplying by 
on both sides, we get
![\[3^{100}+2^{100}=x(3^{96}+2^{96}).\]](http://latex.artofproblemsolving.com/b/b/a/bba3fec961dea84ed97e532bcd451833a01c9cb1.png)
Now let's take a look at the answer choices. We notice that 
, choice 
, can be written as 3^4. Plugging this into out equation above, we get
![\[3^{100}+2^{100} \stackrel{?}{=} 3^4(3^{96}+2^{96}) \Rightarrow 3^{100}+2^{100} \stackrel{?}{=} 3^{100}+3^4*2^{96}.\]](http://latex.artofproblemsolving.com/3/e/5/3e5519a2644138a286af2730918b4f720669a2d8.png)
The right side is larger than the left side because
![\[2^{100} \leq 2^{96}*3^4.\]](http://latex.artofproblemsolving.com/a/d/5/ad5220f2835add12bffe3fb92bb8a484665af1a6.png)
This means that our original value, , must be less than . The only answer that is less than is 
so our answer is 
.
~Nivek
