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Problem 198


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