RMM 2026 Problem 1

Let $n$ be a positive integer. Alice draws a unit area triangle on the board. Then she draws additional triangles by performing $n$ moves in a row. On each move, she chooses a drawn triangle $\Delta$ with no marked points in its interior, marks a point $P$ in its interior, and draws three smaller triangles by joining $P$ to each vertex of $\Delta$ with a segment.

Once these $n$ moves have been performed, Bob chooses three distinct drawn triangles $\Delta_1, \Delta_2,$ and $\Delta_3$ which contain no marked points in their interiors, such that $\Delta_2$ shares one side with $\Delta_1$ and another with $\Delta_3$. In terms of $n$, determine the largest possible constant $c$ such that Bob can guarantee that the sum of the areas of $\Delta_1, \Delta_2,$ and $\Delta_3$ is at least $c$, regardless of Alice’s choices.