CN-TST 2021 Problem 24

Let $n(\ge 2)$ be an integer. $2n^2$ contestants participate in a Chinese chess competition, where any two contestant play exactly once. There may be draws. It is known that (1)If A wins B and B wins C, then A wins C. (2)there are at most $\frac{n^3}{16}$ draws. Proof that it is possible to choose $n^2$ contestants and label them $P_{ij}(1\le i,j\le n)$ , so that for any $i,j,i',j'\in \{1,2,...,n\}$ , if $i<i'$ , then $P_{ij}$ wins $P_{i'j'}$ .