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Problem 770

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Delphi Flip

A and B play a game. A has originally $1$ gram of gold and B has an unlimited amount. Each round goes as follows:

• A chooses and displays, $x$, a nonnegative real number no larger than the amount of gold that A has.
• Either B chooses to TAKE. Then A gives B $x$ grams of gold.
• Or B chooses to GIVE. Then B gives A $x$ grams of gold.

B TAKEs $n$ times and GIVEs $n$ times after which the game finishes.

Define $g(X)$ to be the smallest value of $n$ so that A can guarantee to have at least $X$ grams of gold at the end of the game. You are given $g(1.7)=10$.

Find $g(1.9999)$.

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狡诈的先知

A和B正在玩一个游戏。游戏开始时，A持有$1$克黄金，而B则持有无穷的黄金。在每一轮游戏中：

• A选择并展示一个非负数实数$x$，且$x$不超过A持有的黄金数量。
• B可以选择拿走，此时A必须将$x$克黄金交给B。
• B也可以选择赠予，此时B必须将$x$克黄金交给A。

当B选择了恰好各$n$次拿走和赠予后，游戏结束。

记$g(X)$为最小的$n$，使得A能保证游戏结束时至少持有$X$克黄金。已知$g(1.7)=10$。

求$g(1.9999)$。

译注：本题源自《科学美国人》2001年8月刊同名谜题，大意是赌徒向先知请教接下来数次硬币抛掷的结果，先知虽然能够准确预测未来，但偶尔也会撒谎，赌徒需要决定如何下注以最大化收益。

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