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− | == Solution ==
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− | The best way to start is to just write out some terms.
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− | {| class="wikitable"
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− | |-
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− | | 0 || 1 || 2 || 3 || 4 || 5 || 6
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− | |-
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− | | <math>\quad 1000 \quad</math><font color="white">aa</font> || <math>\quad x \quad</math><font color="white">aaa</font> || <math>1000 - x</math> || <math>2x - 1000</math><font color="white">a</font> || <math>2000 - 3x</math> || <math>5x - 3000</math> || <math>5000 - 8x</math>
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− | |}
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− | It is now apparent that each term can be written as
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− | *<math>n \equiv 0 \pmod{2}\quad\quad F_{n-1}\cdot 1000-F_n\cdot x</math>
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− | *<math>n \equiv 1 \pmod{2}\quad\quad F_{n}\cdot 1000-F_{n-1}\cdot x</math>
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− | where the <math> F_{n}</math> are [[Fibonacci number]]s. This can be proven through induction.
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− | === Solution 1 ===
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− | We can start to write out some of the inequalities now:
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− | #<math>x > 0</math>
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− | #<math>1000 - x > 0 \Longrightarrow x < 1000</math>
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− | #<math>2x - 1000 > 0 \Longrightarrow x > 500</math>
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− | #<math>2000 - 3x > 0 \Longrightarrow x < 666.\overline{6}</math>
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− | #<math>5x - 3000 > 0 \Longrightarrow x > 600</math>
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− | And in general,
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− | *<math>n \equiv 0 \pmod{2}\quad\quad x < \frac{F_{n-1}}{F_n} \cdot 1000</math>
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− | *<math>n \equiv 1 \pmod{2}\quad\quad x > \frac{F_{n-1}}{F_{n}} \cdot 1000</math>
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− | 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):
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− | <math>x < \frac{F_{7}}{F_{8}} \cdot 1000 = \frac{13}{21} \cdot 1000 = 618.\overline{18}</math>
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− | <math>x > \frac{F_{8}}{F_{9}} \cdot 1000 = \frac{21}{34} \cdot 1000 \approx 617.977</math>
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− | Thus the sequence is maximized when <math>x = \boxed{618}.</math>
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| == See also == | | == See also == |
| {{AIME box|year=1998|num-b=7|num-a=9}} | | {{AIME box|year=1998|num-b=7|num-a=9}} |
Revision as of 01:00, 25 August 2024
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?
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.