43 lines
1.2 KiB
Markdown
43 lines
1.2 KiB
Markdown
---
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id: 5900f4331000cf542c50ff45
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title: 'Problem 198: Ambiguous Numbers'
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challengeType: 5
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forumTopicId: 301836
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dashedName: problem-198-ambiguous-numbers
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---
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# --description--
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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 $q > d$.
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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.
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How many ambiguous numbers $x = \frac{p}{q}$, $0 < x < \frac{1}{100}$, are there whose denominator $q$ does not exceed ${10}^8$?
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# --hints--
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`ambiguousNumbers()` should return `52374425`.
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```js
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assert.strictEqual(ambiguousNumbers(), 52374425);
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```
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# --seed--
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## --seed-contents--
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```js
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function ambiguousNumbers() {
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return true;
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}
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ambiguousNumbers();
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```
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# --solutions--
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```js
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// solution required
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```
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