47 lines
1.2 KiB
Markdown
47 lines
1.2 KiB
Markdown
---
|
|
id: 5900f42c1000cf542c50ff3f
|
|
title: 'Problem 192: Best Approximations'
|
|
challengeType: 5
|
|
forumTopicId: 301830
|
|
dashedName: problem-192-best-approximations
|
|
---
|
|
|
|
# --description--
|
|
|
|
Let $x$ be a real number.
|
|
|
|
A best approximation to $x$ for the denominator bound $d$ is a rational number $\frac{r}{s}$ in reduced form, with $s ≤ d$, such that any rational number which is closer to $x$ than $\frac{r}{s}$ has a denominator larger than $d$:
|
|
|
|
$$|\frac{p}{q} - x| < |\frac{r}{s} - x| ⇒ q > d$$
|
|
|
|
For example, the best approximation to $\sqrt{13}$ for the denominator bound $20$ is $\frac{18}{5}$ and the best approximation to $\sqrt{13}$ for the denominator bound $30$ is $\frac{101}{28}$.
|
|
|
|
Find the sum of all denominators of the best approximations to $\sqrt{n}$ for the denominator bound ${10}^{12}$, where $n$ is not a perfect square and $1 < n ≤ 100000$.
|
|
|
|
# --hints--
|
|
|
|
`bestApproximations()` should return `57060635927998344`.
|
|
|
|
```js
|
|
assert.strictEqual(bestApproximations(), 57060635927998344);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function bestApproximations() {
|
|
|
|
return true;
|
|
}
|
|
|
|
bestApproximations();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|