VMO 2020 Problem 7

Given a positive integer $n>1$ . Denote $T$ a set that contains all ordered sets $(x;y;z)$ such that $x,y,z$ are all distinct positive integers and $1\leq x,y,z\leq 2n$ . Also, a set $A$ containing ordered sets $(u;v)$ is called "connected" with $T$ if for every $(x;y;z)\in T$ then $\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing$ . a) Find the number of elements of set $T$ . b) Prove that there exists a set "connected" with $T$ that has exactly $2n(n-1)$ elements. c) Prove that every set "connected" with $T$ has at least $2n(n-1)$ elements.