I have as data some real measures, let's say: 1000, 800, 900, 1100, 900 and I have the Mean Absolute Error (MAE) and Mean Squared Error (MSE) 80 and 20000, but I don't know which are the estimated data. \begin{align} MAE_s(h) &= \frac{1}{n}\sum_{x \in s} |f(x)-h(x)| \\[5pt] MSE_s(h) &= \frac{1}{n}\sum_{x \in s} (f(x)-h(x))^2 \end{align} In the above equations, $f(x)$ is the real measure(1000,800, 900, 1100 and 900) and $h(x)$ is the estimated measure (which is unknown to me), while $n$ is the measure amount (which is 5 in our case).
I have to prove if this situation is possible or not, so I took the MAE formula and I replace the values I knew, getting the equations which you can see below. I used $x,y,z,w,t$ to denote the values that I don't know, which are the the values for the estimated data. \begin{align} 80 &= \frac{1}{5} ((1000-x)+(800-y)+(900-z)+(1100-w)+(900-t)) \\[5pt] 20000 &= \frac{1}{5} ((1000-x)^2+(800-y)^2+(900-z)^2+(1100-w)^2+(900-t)^2) \end{align} I tried different values for $x,y,z,w,t$ until I found one estimated values combination that has the requested MAE and MSE. These values are: 900,800, 900, 800 and 900. So I got something like this: \begin{align} 80 &= \frac{1}{5} ((1000-900)+(800-800)+(900-900)+(1100-800)+(900-900)) \\[5pt] 80 &= 80 \\[5pt] 20000 &= \frac{1}{5} ((1000\!-\!900)^2+(800\!-\!800)^2+(900\!-\!900)^2+(1100\!-\!800)^2+(900\!-\!900)^2) \\[5pt] 20000 &= 20000 \end{align}
So I could prove that the situation was possible, but this doesn't seem a very handy way of doing that because there are a lot of possible combinations.
Is there any other way to check if a situation is possible or not given the MAE, MSE and the real data?