Question 8
Let $n \gt 10$ be an integer, and let $A_1, A_2, \dots, A_n$ be distinct points in the plane such that the distances between the points are pairwise different. Define $f_{10}(j, k)$ to be the 10th smallest of the distances from $A_j$ to $A_1, A_2, \dots, A_k$, excluding $A_j$ if $k \ge j$. Suppose that for all $j$ and $k$ satisfying $11 \le j \le k \le n$, we have $f_{10}(j, j-1) \ge f_{10}(k, j-1)$. Prove that $f_{10}(j, n) \ge \frac{1}{2}f_{10}(n, n)$ for all $j$ in the range $1 \le j \le n-1$.
Iran, Morteza Saghafian
Iran, Morteza Saghafian