2005 AMC 8 Problem 24
Attempt Problem 24 of the 2005 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
24.
A certain calculator has only two keys [] and []. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed “” and you pressed [], it would display “” If you then pressed [], it would display “” Starting with the display “” what is the fewest number of keystrokes you would need to reach “”?
Answer: B
Small hint:
Work backward from to .
Big hint:
For a lower bound, note what eight keystrokes can and cannot reach near .
Solution:
Work backward from . When the number is even, undo by dividing by ; when it is odd, undo by subtracting .
uses reverse steps, so keystrokes are enough.
Eight keystrokes are not enough: the largest value below reachable in keystrokes is obtained by followed by six doublings, giving . The next larger possibilities overshoot to at least , so cannot be reached in keystrokes.
Thus, B is the correct answer.
Problem 24 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
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