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Omni-MATH / Consider sequences a of the form a=(a₁,a₂,…,a₂₀) such that each term a_(i) is either 0 or 1. For each such sequence a, we can produce a sequence b=(b₁,b₂,…,b₂₀), where …

Problem

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problem

Consider sequences aa of the form a=(a1,a2,…,a20)a=\left(a_{1}, a_{2}, \ldots, a_{20}\right) such that each term aia_{i} is either 0 or 1. For each such sequence aa, we can produce a sequence b=(b1,b2,…,b20)b=\left(b_{1}, b_{2}, \ldots, b_{20}\right), where \(b_{i}= \begin{cases}a_{i}+a_{i+1} & i=1 \\ a_{i-1}+a_{i}+a_{i+1} & 1<i<20 \\ a_{i-1}+a_{i} & i=20\end{cases}\). How many sequences bb are there that can be produced by more than one distinct sequence aa?
Plain-text mathematical notation (without MathML)
Consider sequences a of the form a=(a₁,a₂,…,a₂₀) such that each term a_(i) is either 0 or 1. For each such sequence a, we can produce a sequence b=(b₁,b₂,…,b₂₀), where \(b_{i}= \begin{cases}a_{i}+a_{i+1} & i=1 \\ a_{i-1}+a_{i}+a_{i+1} & 1<i<20 \\ a_{i-1}+a_{i} & i=20\end{cases}\). How many sequences b are there that can be produced by more than one distinct sequence a?
Original LaTeX notation
Consider sequences \(a\) of the form \(a=\left(a_{1}, a_{2}, \ldots, a_{20}\right)\) such that each term \(a_{i}\) is either 0 or 1. For each such sequence \(a\), we can produce a sequence \(b=\left(b_{1}, b_{2}, \ldots, b_{20}\right)\), where \(b_{i}= \begin{cases}a_{i}+a_{i+1} & i=1 \\ a_{i-1}+a_{i}+a_{i+1} & 1<i<20 \\ a_{i-1}+a_{i} & i=20\end{cases}\). How many sequences \(b\) are there that can be produced by more than one distinct sequence \(a\)?

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Source and history

task_id
5d28546f-5ee8-5ed6-9c46-043e2af3e15c
task_key
test--5d28546f-5ee8-5ed6-9c46-043e2af3e15c
task_revision_id
1
config / split
/ test

Official source

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