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id	challengeType	title	forumTopicId
5900f4ee1000cf542c510000	5	Problem 385: Ellipses inside triangles	302049

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.

Instructions

Tests

tests:
  - text: <code>euler385()</code> should return 3776957309612154000.
    testString: assert.strictEqual(euler385(), 3776957309612154000);

Challenge Seed

function euler385() {
  // Good luck!
  return true;
}

euler385();

Solution

// solution required

1.5 KiB Raw Blame History