Difference between revisions of "1998 AIME Problems/Problem 8"
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We can start to write out some of the inequalities now: | We can start to write out some of the inequalities now: | ||
− | #<math> | + | #<math>x > 0</math> |
− | #<math>1000 - x | + | #<math>1000 - x > 0 \Longrightarrow x < 1000</math> |
− | #<math> | + | #<math>2x - 1000 > 0 \Longrightarrow x > 500</math> |
− | #<math>3000 - 2x | + | #<math>3000 - 2x > 0 \Longrightarrow x < 666.\overline{6}</math> |
− | #<math>5x - 3000 | + | #<math>5x - 3000 > 0 \Longrightarrow x > 600</math> |
And in general, | And in general, | ||
Line 35: | Line 35: | ||
It is apparent that the bounds are slowly closing in on <math>x</math>, so we can just calculate <math>x</math> for some large value of <math>n</math> (randomly, 10, 11): | It is apparent that the bounds are slowly closing in on <math>x</math>, so we can just calculate <math>x</math> for some large value of <math>n</math> (randomly, 10, 11): | ||
− | <math> | + | <math>x < \frac{F_{9}}{F_{10}} \cdot 1000 = \frac{34}{55} \cdot 1000 = 618.\overline{18}</math> |
− | <math> | + | <math>x > \frac{F_{10}}{F_{11}} \cdot 1000 = \frac{55}{89} \cdot 1000 \approx 617.977</math> |
Thus the sequence is maximized when <math>x = 618</math>. | Thus the sequence is maximized when <math>x = 618</math>. |
Revision as of 15:15, 17 January 2011
Problem
Except for the first two terms, each term of the sequence is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer produces a sequence of maximum length?
Contents
[hide]Solution
The best way to start is to just write out some terms.
0 | 1 | 2 | 3 | 4 | 5 | 6 |
aa | aaa | a |
By now its obvious that the numbers are related to the Fibonacci numbers.
Thus,
Solution 1
We can start to write out some of the inequalities now:
And in general,
It is apparent that the bounds are slowly closing in on , so we can just calculate for some large value of (randomly, 10, 11):
Thus the sequence is maximized when .
Solution 2
It is well known that , so approaches .
See also
1998 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |