Problem 385
Ellipses inside triangles
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.
三角形内的椭圆
对于平面上的任意三角形T，可以证明，完全包含在T内部的面积最大的椭圆是唯一的。
对于给定的n，考虑如下的三角形T：
- T的顶点的坐标均为绝对值≤ n的整数，而且
- T内面积最大的椭圆的焦点1是(√13,0)和(-√13,0)。
记A(n)是所有这样的三角形的面积之和。
例如，当n = 8时，有两个这样的三角形。它们的顶点分别是(-4,-3)、(-4,3)、(8,0)和(4,3)、(4,-3)、(-8,0)，面积均为36。因此A(8) = 36 + 36 = 72。
可以验证，A(10) = 252，A(100) = 34632以及A(1000) = 3529008。
求A(1 000 000 000)。
1椭圆的焦点是椭圆内的两个点A和B，对于椭圆边界上的每一点P，满足AP + PB为常数。