2024 SSMO Relay Round 1 Problems/Problem 3

Problem

Let $T = TNYWR.$ In a circle, there are $T$ people. $T-2$ of them have red shoes, and two of them have blue shoes. First, they will randomly eliminate somebody from the circle. Then, they will randomly eliminate somebody with red shoes from the circle, and the cycle repeats until there is only one person left. If the probability this person has blue shoes is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

Of the $719$ people, $717$ have red shoes. Note that we are guaranteed to remove $359$ of those shoes, on the $2,4,6,\dots$ cycles. Therefore, the question is equivalent to "what is the probability a randomly choosen shoe from $717-359$ red shoes and $2$ blue shoes is blue?" This answer is simply $\frac{2}{2+358} = \frac{1}{180}\implies 1+180 = \boxed{181}.$

~SMO_Team

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2024 SSMO Relay Round 1 Problems/Problem 3

Problem

Solution