2012 AMC 12B 第 14 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。
14.
Bernardo 和 Silvia 玩下面的游戏。选取一个从 到 （含两端）的整数交给 Bernardo。每当 Bernardo 收到一个数时，他把它加倍并把结果交给 Silvia。每当 Silvia 收到一个数时，她给它加上 并把结果交给 Bernardo。最后一个产生小于 的数的人获胜。设 为使 Bernardo 获胜的最小初始数。 的各位数字之和是多少？
Bernardo and Silvia play the following game. An integer between and inclusive, is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds to it and passes the result to Bernardo. The winner is the last person who produces a number less than Let be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of
小提示：
当 Bernardo 加倍后的结果达到 时，他获胜，因为 Silvia 的下一个数会至少为
Bernardo wins on a turn when his doubled result reaches forcing Silvia’s next number to be at least
大提示：
倒推：分别找出经过一、二、三或四轮后，在 Bernardo 的操作后第一次达到 的最小初始数
Work backwards: find the smallest start that first reaches after Bernardo’s move, over one, two, three, or four rounds
解答：
当 Bernardo 在一轮后给出的加倍数 且之前的数都小于 时，他获胜。满足 的最小 是 。
倒推，经过二、三、四轮后导致胜利的最小初始值分别是满足 ，，和 的最小整数，即 ，，和 。不会有超过四轮后才获胜的初始值。
所以 ，各位数字之和为 。
因此正确答案是 A。
Bernardo wins after a round when his doubled number but the previous numbers stayed below The smallest with is
Working backwards, the smallest starting values that lead to a win after two, three, and four rounds are the smallest integers with and namely and No start wins after more than four rounds.
So and the sum of its digits is
Thus, the correct answer is A.
其他年份的第 14 题
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