Problem #437

Equilateral triangles 3
Public 10/30/17 9xp Math 66.7%

Assume a point $D$ inside an equilateral triangle $ABC$ of side length $t$ and with $AD = x$, $BD = y$, $CD = z$, $0 < x \le y \le z$, which is represented as quadruple $(x, y, z, t)$.

If $x, y, z, t$ are all integers and they are coprime, the quadruple $(x, y, z, t)$ is called primitive integer solution. For example, $(57, 65, 73, 112)$ is such a solution. And it’s known that there are infinitely many solutions of primitive integer quadruple $(x, y, z, t)$.

Find all such solutions for $t \le 5000$.

Answer format: [number of solutions],[sum of $x + y + z + t$ of all solutions]

Example: 3,1237 for $t \le 200$
(explanation: there are total 3 solutions $(57, 65, 73, 112)$, $(73, 88, 95, 147)$, $(43, 147, 152, 185)$)

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