Difference between revisions of "2024 SSMO Relay Round 1 Problems/Problem 3"
(Created page with "==Problem== Let <math>T = TNYWR.</math> In a circle, there are <math>T</math> people. <math>T-2</math> of them have red shoes, and two of them have blue shoes. First, they wi...") |
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==Solution== | ==Solution== | ||
| + | Of the <math>719</math> people, <math>717</math> have red shoes. Note that we are guaranteed to remove <math>359</math> of those shoes, on the <math>2,4,6,\dots</math> cycles. Therefore, the question is equivalent to "what is the probability a randomly choosen shoe from <math>717-359</math> red shoes and <math>2</math> blue shoes is blue?" This answer is simply <math>\frac{2}{2+358} = \frac{1}{180}\implies 1+180 = \boxed{181}.</math> | ||
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| + | ~SMO_Team | ||
Latest revision as of 14:43, 10 September 2025
Problem
Let 
In a circle, there are 
people. 
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 
for relatively prime positive integers 
and 
find 
Solution
Of the 
people, 
have red shoes. Note that we are guaranteed to remove 
of those shoes, on the 
cycles. Therefore, the question is equivalent to "what is the probability a randomly choosen shoe from 
red shoes and 
blue shoes is blue?" This answer is simply 
~SMO_Team
