Difference between revisions of "2018 AMC 10A Problems/Problem 14"
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Multiplying by <math>3^{96}+2^{96}</math> on both sides, we get | Multiplying by <math>3^{96}+2^{96}</math> on both sides, we get | ||
<cmath>3^{100}+2^{100}=x(3^{96}+2^{96}).</cmath> | <cmath>3^{100}+2^{100}=x(3^{96}+2^{96}).</cmath> | ||
− | Now let's take a look at the answer choices. We notice that <math>81</math>, choice <math>B</math>, can be written as 3^4. Plugging this into out equation above, we get | + | Now let's take a look at the answer choices. We notice that <math>81</math>, choice <math>B</math>, can be written as <math>3^4</math>. Plugging this into out equation above, we get |
<cmath>3^{100}+2^{100} \stackrel{?}{=} 3^4(3^{96}+2^{96}) \Rightarrow 3^{100}+2^{100} \stackrel{?}{=} 3^{100}+3^4*2^{96}.</cmath> | <cmath>3^{100}+2^{100} \stackrel{?}{=} 3^4(3^{96}+2^{96}) \Rightarrow 3^{100}+2^{100} \stackrel{?}{=} 3^{100}+3^4*2^{96}.</cmath> | ||
The right side is larger than the left side because | The right side is larger than the left side because |
Revision as of 17:20, 8 February 2018
What is the greatest integer less than or equal to
Solution
Let's set this value equal to . We can write Multiplying by on both sides, we get Now let's take a look at the answer choices. We notice that , choice , can be written as . Plugging this into out equation above, we get The right side is larger than the left side because This means that our original value, , must be less than . The only answer that is less than is so our answer is .
~Nivek
Solution 2
Let and . Then our fraction can be written as . Notice that . So , . And our only answer choice less than 81 is
~RegularHexagon
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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All AMC 10 Problems and Solutions |