Claim. The 100 centers all lie inside a rectangle of area at most 400. Proof. Consider two centers at the maximum distance, say $2k$, and orient the plane so that this segment of length $2k$ is horizontal. Then every other center lies within a $2k \times \tfrac{200}{k}$ box, which has area 400. Let $\mathcal{R}$ be such a rectangle, and WLOG suppose its horizontal dimension is at most 20. Divide $\mathcal{R}$ into 10 vertical strips (so that each strip has width at most 2); at least one contains (at least) 10 circle centers. The vertical midline of this strip works.