RMM 2025 Problem 1

Let $n > 10$ be an integer, and let $A_{1},A_{2},\ldots ,A_{n}$ be distinct points in the plane such that the distances between the points are pairwise different. Define $f_{10}(j,k)$ to be the $10^{\mathrm{th}}$ smallest of the distances from $A_{j}$ to $A_{1},A_{2},\ldots ,A_{k}$ , excluding $A_{j}$ if $k\geq j$ . Suppose that for all $j$ and $k$ satisfying $11\leq j\leq k\leq n$ , we have $f_{10}(j,j - 1)\geq f_{10}(k,j - 1)$ . Prove that $f_{10}(j,n)\geq {\frac{1}{2}}f_{10}(n,n)$ for all $j$ in the range $1\leq j\leq n - 1$ .