70 lines
1.4 KiB
Markdown
70 lines
1.4 KiB
Markdown
---
|
||
id: 5900f5131000cf542c510025
|
||
challengeType: 5
|
||
title: 'Problem 422: Sequence of points on a hyperbola'
|
||
forumTopicId: 302092
|
||
---
|
||
|
||
## Description
|
||
<section id='description'>
|
||
Let H be the hyperbola defined by the equation 12x2 + 7xy - 12y2 = 625.
|
||
|
||
Next, define X as the point (7, 1). It can be seen that X is in H.
|
||
|
||
Now we define a sequence of points in H, {Pi : i ≥ 1}, as:
|
||
P1 = (13, 61/4).
|
||
P2 = (-43/6, -4).
|
||
For i > 2, Pi is the unique point in H that is different from Pi-1 and such that line PiPi-1 is parallel to line Pi-2X. It can be shown that Pi is well-defined, and that its coordinates are always rational.
|
||
You are given that P3 = (-19/2, -229/24), P4 = (1267/144, -37/12) and P7 = (17194218091/143327232, 274748766781/1719926784).
|
||
|
||
Find Pn for n = 1114 in the following format:If Pn = (a/b, c/d) where the fractions are in lowest terms and the denominators are positive, then the answer is (a + b + c + d) mod 1 000 000 007.
|
||
|
||
For n = 7, the answer would have been: 806236837.
|
||
</section>
|
||
|
||
## Instructions
|
||
<section id='instructions'>
|
||
|
||
</section>
|
||
|
||
## Tests
|
||
<section id='tests'>
|
||
|
||
```yml
|
||
tests:
|
||
- text: <code>euler422()</code> should return 92060460.
|
||
testString: assert.strictEqual(euler422(), 92060460);
|
||
|
||
```
|
||
|
||
</section>
|
||
|
||
## Challenge Seed
|
||
<section id='challengeSeed'>
|
||
|
||
<div id='js-seed'>
|
||
|
||
```js
|
||
function euler422() {
|
||
|
||
return true;
|
||
}
|
||
|
||
euler422();
|
||
```
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</section>
|
||
|
||
## Solution
|
||
<section id='solution'>
|
||
|
||
```js
|
||
// solution required
|
||
```
|
||
|
||
</section>
|