There have been changes made to the second equation in the pair that will be worth looking at. All values for the solutions must be non-zero positive integers (natural numbers). Please note, all values must be distinct!
According to this equation,
$V_1^2+V_2^2+\cdots+V_k^2 = \left(\frac{2\left(V_{1}+V_{2}+\cdots+V_{k}\right)}{k}-V_{1}\right)^{2}+\left(\frac{2\left(V_{1}+V_{2}+\cdots+V_{k}\right)}{k}-V_{2}\right)^2+\cdots+\left(\frac{2\left(V_{1}+V_{2}\cdots+V_{k}\right)}{k}-V_{k}\right)^{2}$
(Source: Personal observation)
There can be at most only two common solutions to the two equations $V_{1}+V_{2}\cdots+V_{k}=A$ and $V_{1}^{2}+V_{2}^{2}+\cdots +V_{k}^{2}=B$, where $V_{1},V_{2},\cdots V_{k}$ denote different variables whose values I wish to find for a fixed value $A$ and $B$ for each of the two equations.
Is it possible to solve this problem with the least of guesswork application? What if $k$ were to extend into really large numbers, creating about $10^{10}$ variables and above? If not possible (for very large numbers), never mind. Thanks all!
PS Sorry if the tags don't match up with the topic.