RMM 2024 Problem 3

Given a positive integer $n$, a set $\mathcal{S}$ is $n$-admissible if

• each element of $\mathcal{S}$ is an unordered triple of integers in $\{1,2, \ldots, n\}$,
• $|\mathcal{S}|=n-2$, and
• for each $1 \leq k \leq n-2$ and each choice of $k$ distinct $A_{1}, A_{2}, \ldots, A_{k} \in \mathcal{S}$,

$$\left|A_{1} \cup A_{2} \cup \cdots \cup A_{k}\right| \geq k+2$$

Is it true that, for all $n>3$ and for each $n$-admissible set $\mathcal{S}$, there exist pairwise distinct points $P_{1}, \ldots, P_{n}$ in the plane such that the angles of the triangle $P_{i} P_{j} P_{k}$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$ ?

Each problem is worth 7 marks. Time allowed: $4 \frac{1}{2}$ hours.