Problem 198
Ambiguous Numbers
A best approximation to a real number x for the denominator bound d is a rational number r/s (in reduced form) with s ≤ d, so that any rational number p/q which is closer to x than r/s has q > d.
Usually the best approximation to a real number is uniquely determined for all denominator bounds. However, there are some exceptions, e.g. 9/40 has the two best approximations 1/4 and 1/5 for the denominator bound 6. We shall call a real number x ambiguous, if there is at least one denominator bound for which x possesses two best approximations. Clearly, an ambiguous number is necessarily rational.
How many ambiguous numbers x = p/q, 0 < x < 1/100, are there whose denominator q does not exceed 108?
两可数
对于实数x，分母上限为d的最佳逼近，是一个最简分数形式的有理数r/s，其中s ≤ d，使得所有比r/s更接近x的有理数p/q其最简分数形式满足q> d。
通常这样对某个实数带分母上限的最佳逼近是唯一的。然而，也有一些例外，比如对于9/40，分母上限为6的最佳逼近有两个，分别是1/4和1/5。如果对于实数x存在某个分母上限使得它有两个最佳逼近，我们就称x为两可的。显然，一个两可数必须是有理数。
有多少个两可数x = p/q，满足0 < x < 1/100，而且其分母q不超过108？