So many cases, if only there was a shortcut

Note the following is true for positive integers $x_i$ : $(x_i-1)(x_i-2) \geq 0$ Simplifying yields $x_i^2 \geq 3x_i-2$ . Summing this over $i=1, 2, ..., 2017$ gives $x_1^2+x_2^2+\ldots+x_{2017}^2 \geq 3(x_1+x_2+\ldots+x_{2017})-4034 = 9054-4034=5020$ Thus, the expression is greater than or equal to $5020$ wih equality occurring when $x_i \in \{1, 2\}$ . Simplifying this yields $1001$ of the $x_i$ terms are $2$ , and $1016$ are $1$ , which is equality.

Here's how I solved it intuitively, but it's not as nearly as elegant as @Sharky Kesa 's solution.

Convince yourself that, if $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n$ ,

$a_1^2+a_2^2+\cdots+a_n^2<b_1^2+b_2^2+\cdots+b_n^2$

holds true if and only if $a_1,a_2,\cdots,a_n$ have a smaller "spread" than $b_1,b_2,\cdots,b_n$ . This can be more clearly seen if you consider the inequality $a^2+b^2<(a-d)^2+(b+d)^2$ , which holds true for positive $a,b,d$ such that $b>a$ .

(Turns out that this can be expressed more formally: $x_{\text{qm}}^2=x_{\text{am}}^2+\sigma_x^2$ where $x_{\text{qm}}$ is the quadratic mean, $x_{\text{am}}$ the arithmetic mean, and $\sigma_x$ the standard deviation. Thus, for smaller spread (decreasing $\sigma_x$ ) while keeping arithmetic mean constant, the quadratic mean will be smaller.)

With the problem, we want $x_1,x_2,\cdots,x_{2017}$ to be as close as possible together. Since their sum is $3018$ , some of them should be $1$ 's, and the remaining then should be $2$ 's. The only way to arrange this is if $1016$ of them are $1$ 's and the remaining $1001$ are $2$ 's.

Thus, the minimum value is $1016\cdot 1^2+1001\cdot 2^2=5020$ .

For positive reals, we see the equality holds when each variable is around 1.49....
If we constraint ourselves to positive integers, we see that each x must be in the vicinity of the above number.......that means x can be 1 or 2
Assuming there are "k" ones and "2017-k" twos, we can add up and find that k equals 1016........
Hence the minimum occurs and comes out to be 5020..........!!