I was working on writing some problems for a contest, and I wrote the following system of equations:
\begin{align*} x^2+yz&=259,\\ y^2+zx&=217,\\ z^2+xy&=203. \end{align*}
Of course, this was written "backwards." I was interested in this system, chose a solution $(x,y,z)$, and then plugged in the values to create the problem.
Unfortunately, I can't find a solution that goes from this system to the ordered triple $(x,y,z)$. The first thing I tried was subtracting the equations in pairs and factoring to get the new system
\begin{align*} (x-y)(x+y-z)&=42,\\ (y-z)(y+z-x)&=14,\\ (z-x)(z+x-y)&=56.\\ \end{align*}
Although this system seems like it should be easier to work with, I can't make any progress, and am beginning to doubt if its even an improvement on the original. Adding all 3 equations and multiplying by 2 gives
$$(x+y)^2+(y+z)^2+(z+x)^2=1358,$$
which also seems useful, but I am still unable to use to solve the system.
Also, multiplying each equation by $x$, $y$, or $z$ and summing gives
$$x^3+y^3+z^3+3xyz=259x+217y+203z,$$
which might be promising since the polynomial $x^2+y^3+z^3-3xyz$ factors nicely.
Since the system is relatively "nice" and symmetric, I was expecting to find a simple solution, and might just be missing something obvious. Any help is much appreciated.