Problem: Each point in the plane is assigned a real number.
Suppose that for any nondegenerate triangle, the number at its incenter
is the arithmetic mean of the three numbers at its vertices.
Prove that all points in the plane were assigned the same number.

\end{enumerate}

Solution: In this solution, we denote points by capital letters
and the associated values by lowercase letters
(e.g.\ $p$ denotes the value of our function at $P$).

\begin{claim*}
  If $ABCD$ is an isosceles trapezoid, then $a+c=b+d$.
\end{claim*}
\begin{proof}
  Indeed, suppose WLOG that rays $BA$ and $CD$ meet at $X$.
  Then triangles $XAC$ and $XBD$ share an incircle, proving the claim.
\end{proof}

Now, given any two points $A$ and $B$, construct regular pentagon $ABCDE$.
We have \[ a+c=b+d=c+e=d+a=e+b \implies a=b=c=d=e \]
as desired.