Given $n$ points in general position in the plane, let $P_n$ be the maximum proportion of the $\binom{n}{3}$ triangles with three acute angles. What is the limit $\lim\limits_{n \rightarrow \infty} P_n$?
An upper bound of $\frac{7}{10}$ is shown in the proof of the sixth question of the 1970 edition of the International Mathematical Olympiad.
In the opposite direction, we can establish a trivial lower bound of $5/9$ by spreading the points at $A$ so they are on a circle, centre $B$; the points at $B$ so they are on a circle, centre $C$, and the points at $C$ are on a circle centre $A$. If the points at $A$ are on an arc of width $N^{-13}$, those at $B$ on an arc of width $N{-9}$ and those at $C$ on an arc of width $N^{-11}$; and further the angle at $B$ is $\pi/2-N^{-7}$, then all triangles $A_iA_jB_k, B_iB_jC_k$ and $C_iC_jA_k$ are acute.
