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2025 SSMO Relay Round 3 Problems

Problem 1

Michael, Nico, and Jackson are splitting a $100$ dollar bill. Michael's favorite number is $3,$ so he insists on paying a positive amount of dollars that is a multiple of $3$. Nico and Jackson are both scared of even numbers, so they each insist on paying an odd number of dollars. In how many ways can they pay the bill?

Solution

Problem 2

Let $T = TNYWR.$ In triangle $ABC,$ angle $C$ is a right angle. If the median from $C$ has length $T,$ find the maximum possible area of triangle $ABC$.

Solution

Problem 3

Let $T = TNYWR.$ The numbers $1$ and $2$ are written on a board. Every second, if the numbers currently on the board are $x$ and $y,$ the number $x+y$ is written, and one of $x$ or $y$ is erased. After a finite number of seconds, the larger number on the board is $\left\lfloor\sqrt{T}\right\rfloor$. How many possible values can the smaller number take?

Solution

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