2025 SSMO Tiebreaker Round Problems
Problem 1
In a triangular grid, each cell must contain exactly one number such that:
- Each cell in the bottom row contains either
or 
. - Each cell not in the bottom row contains the sum of the two numbers in the cells directly below it.
or 
.Determine the number in the topmost cell.
![\[\begin{array}{cc} \boxed{\phantom{00}} \\ \boxed{\phantom{00}} \text{ } \boxed{25} \\ \boxed{12}\text{ } \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}} \\ \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \\ \boxed{\phantom{00}} \text{ }\boxed{\phantom{00}}\text{ }\boxed{\phantom{00}}\text{ } \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}} \\ \boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \text{ }\boxed{\phantom{00}}\text{ } \boxed{\phantom{00}} \text{ }\boxed{\phantom{00}} \\ \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}}\text{ }\boxed{\phantom{00}} \text{ }\boxed{\phantom{00}}\text{ } \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}}\text{ } \boxed{\phantom{00}} \end{array}\]](http://latex.artofproblemsolving.com/1/7/1/171b2d9a5a2d751b57f86dee0a58a703477a4808.png)
Problem 2
Determine the largest positive integer 
satisfying ![\[\frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+2026} - \sqrt{n+2025}} > 2.\]](http://latex.artofproblemsolving.com/7/b/1/7b1de01cd9a5799c64592816146e4fda12106aae.png)
Problem 3
Find the number of functions 
such that 
for all 
.
