US-TST 2024 Problem 4

Find all integers $n\geq 2$ for which there exists a sequence of $2n$ pairwise distinct points $(P_{1},\ldots ,P_{n},Q_{1},\ldots ,Q_{n})$ in the plane satisfying the following four conditions:

(i) no three of the $2n$ points are collinear;

(ii) $P_{i}P_{i + 1}\geq 1$ for all $i = 1,2,\ldots ,n$ , where $P_{n + 1} = P_{1}$

(iii) $Q_{i}Q_{i + 1}\geq 1$ for all $i = 1,2,\ldots ,n$ , where $Q_{n + 1} = Q_{1}$ ; and

(iv) $P_{i}Q_{j}\leq 1$ for all $i = 1,2,\ldots ,n$ and $j = 1,2,\ldots ,n$