HMMT 2003 C1
- Combinatorics
- Answer
You have 2003 switches, numbered from 1 to 2003, arranged in a circle. Initially, each switch is either ON or OFF, and all configurations of switches are equally likely. You perform the following operation: for each switch , if the two switches next to were initially in the same position, then you set to ON; otherwise, you set to OFF. What is the probability that all switches will now be ON?
Dataset text NuminaMath olympiads. Quoted with source.
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\frac{1}{2^{2002}}
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