Difference between revisions of "2018 AMC 10A Problems/Problem 14"
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{{AMC10 box|year=2018|ab=A|num-b=13|num-a=15}} | {{AMC10 box|year=2018|ab=A|num-b=13|num-a=15}} | ||
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− | Let <math>x=\frac{3^100+2^100}{3^96+2^96}</math>. Multiply both sides by <math>{3^96+2^96}</math>, and expand. Rearranging the terms, we get <math>3^96{3^4-x}+2^96{2^4-x}=0</math>. The left side is strictly decreasing, and it is negative when <math>x=81</math>. Therefore the answer must be less than <math>81</math>, therefore the answer is <math>\boxed{(A)}</math>. | + | Let <math>x=\frac{3^{100}+2^{100}}{3^{96}+2^{96}}</math>. Multiply both sides by <math>{3^{96}+2^{96}}</math>, and expand. Rearranging the terms, we get <math>3^{96}{3^4-x}+2^{96}{2^4-x}=0</math>. The left side is strictly decreasing, and it is negative when <math>x=81</math>. Therefore the answer must be less than <math>81</math>, therefore the answer is <math>\boxed{(A)}</math>. |
Revision as of 18:45, 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 |
Solution 3
Let . Multiply both sides by , and expand. Rearranging the terms, we get . The left side is strictly decreasing, and it is negative when . Therefore the answer must be less than , therefore the answer is .