2025 SSMO Relay Round 5 Problems
Problem 1
The numbers 
and 
are written on a blackboard. Every second, if the number 
is currently written on the blackboard, it is replaced by 
with probability 
and by 
with probability 
. For example, after one second, the two numbers on the blackboard may be 
and 
. Given that the expected positive difference between the two numbers on the blackboard after six seconds is 
the value of can be expressed as 
where 
and 
are positive integers. Find 
.
Problem 2
Let 
Ethan places 
red cards and blue cards in a row so that the leftmost card is red and every second card is red. He performs a sequence of operations called replacements: in each replacement, he randomly selects a card other than the leftmost one, discards it, and moves the leftmost card to the discarded card’s position. This process is repeated until only one card remains. The probability that the final remaining card is red is 
for relatively prime positive integers 
and . Find 
.
Problem 3
Let How many positive integers satisfy ![\[\operatorname{lcm}(\operatorname{gcd}(n, n^2 + T^2), \operatorname{gcd}(T, n^2 + T^2)) = \operatorname{lcm}(n, T)?\]](http://latex.artofproblemsolving.com/b/7/8/b7888a8ca4bef71324f6f2efbb65aadb0c2d7cbb.png)
