54 lines
1.3 KiB
Markdown
54 lines
1.3 KiB
Markdown
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---
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id: 5900f4ee1000cf542c510000
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title: 'Problem 385: Ellipses inside triangles'
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challengeType: 5
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forumTopicId: 302049
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dashedName: problem-385-ellipses-inside-triangles
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---
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# --description--
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For any triangle T in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside T.
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For a given n, consider triangles T such that:
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- the vertices of T have integer coordinates with absolute value ≤ n, and
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- the foci1 of the largest-area ellipse inside T are (√13,0) and (-√13,0).
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Let A(n) be the sum of the areas of all such triangles.
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For example, if n = 8, there are two such triangles. Their vertices are (-4,-3),(-4,3),(8,0) and (4,3),(4,-3),(-8,0), and the area of each triangle is 36. Thus A(8) = 36 + 36 = 72.
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It can be verified that A(10) = 252, A(100) = 34632 and A(1000) = 3529008.
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Find A(1 000 000 000).
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1The foci (plural of focus) of an ellipse are two points A and B such that for every point P on the boundary of the ellipse, AP + PB is constant.
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# --hints--
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`euler385()` should return 3776957309612154000.
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```js
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assert.strictEqual(euler385(), 3776957309612154000);
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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 euler385() {
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return true;
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}
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euler385();
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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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