Let S be a triplet set of integers $(x, y, z)$ satisfying the equation $∣x∣ + ∣y∣ + ∣z∣ = 2021$. If we define $g (x, y, z) = x + y + z$ for every real number $x, y$, and $z$, calculate the value of $\sum_{(x,y,z) \in S} g(x,y,z).$
Previously I have looked for the x y and z values but the results do not match, is this problem using number theory?
A MASSIVE HINT
@Lulu has pointed out that solutions occur in pairs $(x,y,z)$ and $(-x,-y,-z)$.
Consider for example the pair of solutions $(3,-4,2014)$ and $(-3,4,-2014)$. What is the value of $(3-4+2014)+(-3+4-2014)$?
Do you now see what the point of the hint was?