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 $1000$ .

Solution

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2020 CIME I (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2020 CIME I Problems/Problem 8

Problem 8

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