US-EGMO-TST 2026 Problem 5

Let $n$ be a positive integer. In the infinite lattice $\mathbb{Z}^2$, $n$ points are colored red while the rest are colored blue. Each red point is labeled with the distance to the nearest blue point in the same row or column. Find the smallest real number $\alpha$ for which the sum of all labels does not exceed $100n^\alpha$, independent of $n$ and the placement of the red points.

(Note: A row is the set of points with a given $y$-coordinate, and a column is the set of points with a given $x$-coordinate.)