2022 SSMO Relay Round 4 Problems
Problem 1
On any given day, there is a 
chance that a robot will find a new organism, a 
chance it will find an already discovered organism, and a 
chance that it will find nothing. Given that it has found a new organism, there is a 
chance it will correctly determine that it is a new organism, and given that it has found an already discovered organism, there is a 
chance that it will correctly determine that it has already been discovered. The expected number of days that the robot will take to report that it has found a new organism (regardless of whether it actually has) can be expressed as 
where 
and 
are relatively prime positive integers. Find 
Problem 2
The roots of 
are 
Let 
be a polynomial with roots 
If ![\[S=\sum_{n=1}^{T}(-1)^{n}g_n(5),\]](http://latex.artofproblemsolving.com/d/4/5/d4599d33d3a9b9d0c86103b46aa54cff389988a7.png)
find the remainder when 
is divided by 1000.
Problem 3
Let 
TNYWR. If 
, 
, and ![\[f(n+2)=12f(n+1)-20f(n)\]](http://latex.artofproblemsolving.com/e/5/8/e5803550b5cb87d23083ccc09f35d30c5f8052ca.png)
for all positive integers , find the remainder when 
is divided by 
