1.3 KiB
1.3 KiB
id | title | challengeType | forumTopicId | dashedName |
---|---|---|---|---|
5900f4ee1000cf542c510000 | Problem 385: Ellipses inside triangles | 5 | 302049 | problem-385-ellipses-inside-triangles |
--description--
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.
For a given n, consider triangles T such that:
- the vertices of T have integer coordinates with absolute value ≤ n, and
- the foci1 of the largest-area ellipse inside T are (√13,0) and (-√13,0).
Let A(n) be the sum of the areas of all such triangles.
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.
It can be verified that A(10) = 252, A(100) = 34632 and A(1000) = 3529008.
Find A(1 000 000 000).
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.
--hints--
euler385()
should return 3776957309612154000.
assert.strictEqual(euler385(), 3776957309612154000);
--seed--
--seed-contents--
function euler385() {
return true;
}
euler385();
--solutions--
// solution required