# Difference between revisions of "2020 CIME I Problems/Problem 8"

## Problem 8

A person has been declared the first to inhabit a certain planet on day $N=0$. For each positive integer $N>0$, if there is a positive number of people on the planet, then either one of the following three occurs, each with probability $\frac{1}{3}$:

(i) the population stays the same;
(ii) the population increases by $2^N$; or
(iii) the population decreases by $2^{N-1}$. (If there are no greater than $2^{N-1}$ people on the planet, the population drops to zero, and the process terminates.)

The probability that at some point there are exactly $2^{20}+2^{19}+2^{10}+2^9+1$ people on the planet can be written as $\frac{p}{3^q}$, where $p$ and $q$ are positive integers such that $p$ isn't divisible by $3$. Find the remainder when $p+q$ is divided by $1000$.