Maximizing the sum of the squares of numbers whose sum is constant

Question

I wonder how one goes about to find the maximum of $\sum v_i^2$, the $v_i$'s being positive integers whose sum $\sum_i v_i$ is fixed.

Answer 1

If there are two numbers with $1\lt v_j\le v_k$, you can decrease $v_j$ and increase $v_k$ to increase the sum of squares. Thus at most one of the $v_i$ is greater than $1$.

Answer 2

Imagine that you're aiming to cover as much of the $\sum_i v_i$ square as possible:

The bigger the largest inner square, the closer it gets to covering more of the background square. The maximum sum of squares is reached when all but one of the $v_i$ is at the specified minimum - in this case, $1$.