Difference between revisions of "2016 AIME II Problems/Problem 9"
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== Solution 2== | == Solution 2== | ||
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+ | We have <math>a_k=r^{k-1}</math> and <math>b_k=(k-1)d</math>. First, <math>b_{k-1}<c_{k-1}=100</math> implies <math>d<100</math>, so <math>b_{k+1}<300</math>. | ||
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+ | It follows that <math>a_{k+1}=1000-b_{k+1}>700</math>, i.e., <math>700 < r^k < 1000</math>. Dividing through by <math>r^2</math> we get <cmath>\frac{700}{r^2} < r^{j} < \frac{1000}{r^2}</cmath>where <math>j=k-2</math>. Since <math>k</math> is atleast <math>3</math> we get <math>r^3\le r^k <1000</math>, i.e. <math>r<10</math>. Let's make a table: | ||
+ | <cmath>\begin{array}[b]{ c c c c c c c c c } | ||
+ | r & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\[2ex] | ||
+ | \left\lfloor\frac{700}{r^2}\right\rfloor & 175 & 77 & 43 & 28 & 19 & 14 & 10 & 8\\[2ex] | ||
+ | \left\lfloor\frac{1000}{r^2}\right\rfloor & 250 & 111 & 63 & 40 & 28 & 20 & 16 & 12 | ||
+ | \end{array} </cmath> | ||
+ | The only admissible <math>r^j</math> values are <math>\{3^4, 9\}</math>. | ||
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+ | Since <math>100=c_{j+1}=r^j+jd+1</math>, we must have <math>j\mid 99-r^j</math>. | ||
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+ | Note that <math>99-3^4=18</math>, which is not a multiple of <math>4</math>, which leaves <math>r^j=9^1</math>. We check: <math>a_j=9</math> implies <math>b_j=91</math>, i.e. <math>d=90</math>, so <math>b_{j+2}=271</math> and <math>a_{j+2}=729</math> and <math>c_{j+2}=1000</math>! So it works! Then <math>c_{j+1}=9^2+181=262</math>. | ||
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+ | == Solution 3== | ||
Using the same reasoning (<math>100</math> isn't very big), we can guess which terms will work. The first case is <math>k=3</math>, so we assume the second and fourth terms of <math>c</math> are <math>100</math> and <math>1000</math>. We let <math>r</math> be the common ratio of the geometric sequence and write the arithmetic relationships in terms of <math>r</math>. | Using the same reasoning (<math>100</math> isn't very big), we can guess which terms will work. The first case is <math>k=3</math>, so we assume the second and fourth terms of <math>c</math> are <math>100</math> and <math>1000</math>. We let <math>r</math> be the common ratio of the geometric sequence and write the arithmetic relationships in terms of <math>r</math>. |
Revision as of 20:59, 23 November 2021
Contents
Problem
The sequences of positive integers and are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let . There is an integer such that and . Find .
Solution 1
Since all the terms of the sequences are integers, and 100 isn't very big, we should just try out the possibilities for . When we get to and , we have and , which works, therefore, the answer is .
Solution 2
We have and . First, implies , so .
It follows that , i.e., . Dividing through by we get where . Since is atleast we get , i.e. . Let's make a table: The only admissible values are .
Since , we must have .
Note that , which is not a multiple of , which leaves . We check: implies , i.e. , so and and ! So it works! Then .
Solution 3
Using the same reasoning ( isn't very big), we can guess which terms will work. The first case is , so we assume the second and fourth terms of are and . We let be the common ratio of the geometric sequence and write the arithmetic relationships in terms of .
The common difference is , and so we can equate: . Moving all the terms to one side and the constants to the other yields
, or . Simply listing out the factors of shows that the only factor less than a square that works is . Thus and we solve from there to get .
Solution by rocketscience
Solution 3 (More Robust Bash)
The reason for bashing in this context can also be justified by the fact 100 isn't very big.
Let the common difference for the arithmetic sequence be , and the common ratio for the geometric sequence be . The sequences are now , and . We can now write the given two equations as the following:
Take the difference between the two equations to get . Since 900 is divisible by 4, we can tell is even and is odd. Let , , where and are positive integers. Substitute variables and divide by 4 to get:
Because very small integers for yield very big results, we can bash through all cases of . Here, we set an upper bound for by setting as 3. After trying values, we find that , so . Testing out yields the correct answer of . Note that even if this answer were associated with another b value like , the value of can still only be 3 for all of the cases.
-Dankster42
See also
2016 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.