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Problem 562

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Maximal perimeter

Construct triangle ABC such that:

• Vertices A, B and C are lattice points inside or on the circle of radius r centered at the origin;
• the triangle contains no other lattice point inside or on its edges;
• the perimeter is maximum.

Let R be the circumradius of triangle ABC and T(r) = R/r.
For r = 5, one possible triangle has vertices (-4,-3), (4,2) and (1,0) with perimeter $\sqrt{13}+\sqrt{34}+\sqrt{89}$ and circumradius R = $\sqrt {\frac {19669} 2 }$, so T(5) =$\sqrt {\frac {19669} {50} }$.
You are given T(10) ~ 97.26729 and T(100) ~ 9157.64707.

Find T(10⁷). Give your answer rounded to the nearest integer.

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最大周长

构造满足如下条件的三角形ABC：

• 顶点A，B和C都是在以原点为圆心，r为半径的圆内或圆周上的格点；
• 三角形内或边上没有其它的格点；
• 周长最大。

记R是三角形ABC的外接圆半径，并记T(r) = R/r。
对于r = 5，一种可能的方案是以(-4,-3)，(4,2)和(1,0)为顶点的三角形，周长为$\sqrt{13}+\sqrt{34}+\sqrt{89}$，外接圆半径为R = $\sqrt {\frac {19669} 2 }$，因此T(5) =$\sqrt {\frac {19669} {50} }$。
已知T(10) ~ 97.26729以及T(100) ~ 9157.64707。

求T(10⁷)，将你的答案四舍五入到最接近的整数。

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