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id	title	challengeType	forumTopicId	dashedName
5900f4331000cf542c50ff45	Problem 198: Ambiguous Numbers	5	301836	problem-198-ambiguous-numbers

--description--

A best approximation to a real number x for the denominator bound d is a rational number \frac{r}{s} (in reduced form) with s ≤ d, so that any rational number \frac{p}{q} which is closer to x than \frac{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. \frac{9}{40} has the two best approximations \frac{1}{4} and \frac{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 = \frac{p}{q}, 0 < x < \frac{1}{100}, are there whose denominator q does not exceed {10}^8?

--hints--

ambiguousNumbers() should return 52374425.

assert.strictEqual(ambiguousNumbers(), 52374425);

--seed--

--seed-contents--

function ambiguousNumbers() {

  return true;
}

ambiguousNumbers();

--solutions--

// solution required

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