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id	title	challengeType	forumTopicId	dashedName
5900f4241000cf542c50ff37	Problem 184: Triangles containing the origin	5	301820	problem-184-triangles-containing-the-origin

--description--

Consider the set I_r of points (x,y) with integer coordinates in the interior of the circle with radius r, centered at the origin, i.e. x^2 + y^2 < r^2.

For a radius of 2, I_2 contains the nine points (0,0), (1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1) and (1,-1). There are eight triangles having all three vertices in I_2 which contain the origin in the interior. Two of them are shown below, the others are obtained from these by rotation.

circle with radius 2, centered at the origin, with nine marked points and two triangles - (-1,0), (0,1), (1,-1) and (-1,1), (0,-1), (1,1)

For a radius of 3, there are 360 triangles containing the origin in the interior and having all vertices in I_3 and for I_5 the number is 10600.

How many triangles are there containing the origin in the interior and having all three vertices in I_{105}?

--hints--

trianglesConttainingOrigin() should return 1725323624056.

assert.strictEqual(trianglesConttainingOrigin(), 1725323624056);

--seed--

--seed-contents--

function trianglesContainingOrigin() {

  return true;
}

trianglesContainingOrigin();

--solutions--

// solution required

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