INMO 2023 Problem 5

Euler marks $n$ different points in the Euclidean plane. For each pair of marked points, Gauss writes down the number $\left\lfloor\log _{2} d\right\rfloor$ where $d$ is the distance between the two points. Prove that Gauss writes down less than $2 n$ distinct values.

Note: For any $d>0,\left\lfloor\log _{2} d\right\rfloor$ is the unique integer $k$ such that $2^{k} \leq d<2^{k+1}$.