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 $1000.$ $[asy] size(5cm); fill(circle((0,0),10),black); fill(circle((0,0),9),white); fill(circle((0,0),8),black); fill(circle((0,0),7),white); fill(circle((0,0),3),black); fill(circle((0,0),2),white); fill(circle((0,0),1),black); dot((0,6)); dot((0,5)); dot((0,4)); [/asy]$

Solution

Note that $\begin{align*} k &= 2023^2-2022^2+2021^2-2019^2\ldots+3^2-2^1+1^2\\ &= (2023-2022)(2023+2022)+\ldots+(3-2)(3+2)+1^2\\ &= (2023+2022)+(2021+2019)+\ldots+(3+2)+1\\ &= \frac{2023\cdot2024}{2}\implies\\ k\pmod{1000}&\equiv\boxed{276}. \end{align*}$

~pinkpig

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2023 WSMO Team Round Problems/Problem 3

Problem

Solution