Question 4
Let $n \ge 3$ be an integer. A 3-element subset $A$ of the set $M = \{1, 2, 3, \dots, n\}$ will be called interesting if there exists a set $B$ of positive integers so that:
(1) if $a \in A, b \in A, a \neq b$, then $(a+b) \in B$;
(2) if $x, y, z \in B, x \lt y \lt z$, then $2 \cdot y = x + z$.
Denote $A_n$ the number of the interesting sets.
a) Compute $A_7$.
b) Prove that $A_{91} \gt 2024$.
(1) if $a \in A, b \in A, a \neq b$, then $(a+b) \in B$;
(2) if $x, y, z \in B, x \lt y \lt z$, then $2 \cdot y = x + z$.
Denote $A_n$ the number of the interesting sets.
a) Compute $A_7$.
b) Prove that $A_{91} \gt 2024$.