Problem 198Published on Saturday, 14th June 2008, 02:00 am; Solved by 706; Difficulty rating: 80%
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?