Difference between revisions of "2020 CIME I Problems/Problem 8"
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− | A person has been declared the first to inhabit a certain planet on day <math>N=0</math>. For each positive integer <math>N | + | A person has been declared the first to inhabit a certain planet on day <math>N=0</math>. For each positive integer <math>N>0</math>, if there is a positive number of people on the planet, then either one of the following three occurs, each with probability <math>\frac{1}{3}</math>: |
:(i) the population stays the same; | :(i) the population stays the same; |
Latest revision as of 17:00, 31 August 2020
Problem 8
A person has been declared the first to inhabit a certain planet on day . For each positive integer , if there is a positive number of people on the planet, then either one of the following three occurs, each with probability :
- (i) the population stays the same;
- (ii) the population increases by ; or
- (iii) the population decreases by . (If there are no greater than people on the planet, the population drops to zero, and the process terminates.)
The probability that at some point there are exactly people on the planet can be written as , where and are positive integers such that isn't divisible by . Find the remainder when is divided by .
Solution
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2020 CIME I (Problems • Answer Key • Resources) | ||
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