2025 SSMO Tiebreaker Round Problems/Problem 1

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Problem

In a triangular grid, each cell must contain exactly one number such that:

  • Each cell in the bottom row contains either $0$ or $1$.
  • Each cell not in the bottom row contains the sum of the two numbers in the cells directly below it.

Determine the number in the topmost cell.

\[\begin{array}{cc} \boxed{\phantom{00}} \\ \boxed{\phantom{00}} \text{ } \boxed{25} \\ \boxed{12}\text{ } \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}} \\ \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \\ \boxed{\phantom{00}} \text{ }\boxed{\phantom{00}}\text{ }\boxed{\phantom{00}}\text{ } \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}} \\ \boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \text{ }\boxed{\phantom{00}}\text{ } \boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \\ \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}}\text{ }\boxed{\phantom{00}} \text{ }\boxed{\phantom{00}}\text{ } \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}}  \end{array}\]

Solution