2005 AMC 8 第 24 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
24.
某计算器只有两个键 [] 和 []。按下其中一个键时，计算器会自动显示结果。例如，如果计算器原来显示“”，按 [] 后会显示“”。如果接着按 []，会显示“”。从显示“”开始，要达到“”最少需要按多少次键？
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 “”?
小提示：
从 反向推回
Work backward from to .
大提示：
为了给出下界，注意八次按键在 附近能到达和不能到达什么
For a lower bound, note what eight keystrokes can and cannot reach near .
解答：
从 反向操作。数为偶数时，用除以 撤销 ；数为奇数时，用减去 撤销 。
使用 个反向步骤，所以 次按键足够。
对于目标 ， 次按键不够：小于目标的最大可达值由先 ，再六次加倍得到，即 。更大的可能会至少达到 ，所以 不能在 次按键内到达。
所以正确答案是 B。
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.
其他年份的第 24 题
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