Suppose $x$, $y$, and $z$ are integers that satisfy the system equations

Question

Suppose $x$, $y$, and $z$ are integers that satisfy the system equations :

$x^2y$+ $y^2z$ + $z^2x$ $= 2186$

$y^2x$+ $z^2y$ + $x^2z$ $= 2188$

What is $x^2+y^2+z^2$ ?

Answer 1

Subtracting the first equation from the second, we have: $$ y^2(x - z) + z^2 (y-x) + x^2(z-y) = 2$$ $$ \implies \qquad (x - z)(y-x) (y - z) = 2 \tag {3}$$

Now considering the possible integer factors of 2, could you solve for possible $(x, y, z)$?

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ADDITIONAL HINTS

a) The difference between these integers can be only among $\pm1, \pm2$, and should satisfy (3). So assume $x=a$ and $x <y<z$, can you write $y$ and $z$ in terms of $a$? What if the ordering is different - are there other solutions?

b) To find $a$, substitute solutions obtained from above into your first equation.