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Problem #412

Symmetric Equation I
 Public ★(x11) 07/28/17 by Philippe_57721 7xp Programming 47.8%

Here is the smallest known solution in positive integers for the Diophantine equation $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} = 4$$
$x= 43736[\dots]772036$
‌​ $y= 368751[\dots]055579‌​$
$z=1544768[\dots]27799‌​9$
(The $\dots$ represent 70 digits !)

If we allow one variable to be negative, we can find smaller solutions.

For example, (x,y,z) = {-1,4,11} and {-5,9,11} are solutions of the above equation.

Find the first solution to
$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} = 38 \textrm{ with } x \lt 0 \lt y \lt z \textrm{ (coprime integers)}$

[My timing: 15 sec]

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