Let
![n \geq 4](/media/m/4/f/2/4f2c159d62a3247dd33ee8d022ae9f37.png)
be a fixed positive integer. Given a set
![S = \{P_1, P_2, \ldots, P_n\}](/media/m/f/f/9/ff9397ba82d11607a6e0c7cb655f2e07.png)
of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
points in the plane such that no three are collinear and no four concyclic, let
![1 \leq t \leq n,](/media/m/a/3/f/a3f7d3c3fc200fc204cf544e2eba3c7a.png)
be the number of circles
![P_iP_jP_k](/media/m/6/2/5/6259780a96e6682e08d8d8006a7fa1c4.png)
that contain
![P_t](/media/m/a/8/5/a858e0aa0d1d4e288d1613f00827a516.png)
in their interior, and let
![m(S) = \sum^n_{i=1} a_i.](/media/m/9/a/b/9abcd50d89144782ff308cc6b09b76bc.png)
Prove that there exists a positive integer
![f(n),](/media/m/8/b/a/8ba75e04f74d8d0dbbc5921c86e3b144.png)
depending only on
![n,](/media/m/5/f/2/5f26ebab144fee216fbc733cb1fa2f2b.png)
such that the points of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
are the vertices of a convex polygon if and only if
%V0
Let $n \geq 4$ be a fixed positive integer. Given a set $S = \{P_1, P_2, \ldots, P_n\}$ of $n$ points in the plane such that no three are collinear and no four concyclic, let $a_t,$ $1 \leq t \leq n,$ be the number of circles $P_iP_jP_k$ that contain $P_t$ in their interior, and let $m(S) = \sum^n_{i=1} a_i.$ Prove that there exists a positive integer $f(n),$ depending only on $n,$ such that the points of $S$ are the vertices of a convex polygon if and only if $m(S) = f(n).$