2023 WSMO Team Round Problems/Problem 3

Problem

In the figure below, there are $1023$ total circles. The area between circles alternate between shaded and non-shaded. If the area of the shaded region is $k\pi,$ find the remainder when $k$ is divided by $1000.$ $[asy] size(5cm); fill(circle((0,0),10),black); fill(circle((0,0),9),white); fill(circle((0,0),8),black); fill(circle((0,0),7),white); fill(circle((0,0),3),black); fill(circle((0,0),2),white); fill(circle((0,0),1),black); dot((0,6)); dot((0,5)); dot((0,4)); [/asy]$

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2023 WSMO Team Round Problems/Problem 3

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