Puzzles are really fun!

Use the symmetry to simplify. You can subtract any number from all circles so subtract 500. There is reflection symmetry about two axis. The solution will be symmetric about one and anti-symmetric about another. The centre number must therefore be zero and the others x,-x,-x,x giving just one equation to solve (500+x)/3 = -x, x = 125. Now add the 500 back on to get the answer 125+500=625.

Let $a,b,c,d,e$ be the missing numbers, from top to bottom, reading from left to right. (That is, $c$ is the center one, $a,b$ are adjacent to $0$ ...) We obtain the five equations: $\begin{cases} 3a = 0+c+d \\ 3b = 0+c+e \\ 6c = 0+a+b+d+e+1000 \\ 3d = a+c+1000 \\ 3e = b+c+1000 \end{cases}$ Solving the system above gives $(a,b,c,d,e) = (375,375,500,625,625)$ , thus the largest is $\boxed{625}$ . Solving the system should be simple enough. (I used the first and fourth equations to obtain $a+d$ in $c$ , as well as $b+e$ with the second and fifth, so substituting to the third equation gives $c$ . I then plug them back into the first and fourth equations to get $a,d$ , and second and fifth to get $b,e$ .)