benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / For any odd positive integer n, let r(n) be the odd positive integer such that the binary representation of r(n) is the binary representation of n written backwards. For example, …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

For any odd positive integer nn, let r(n)r(n) be the odd positive integer such that the binary representation of r(n)r(n) is the binary representation of nn written backwards. For example, r(2023)=r(111111001112)=111001111112=1855r(2023)=r\left(11111100111_{2}\right)=11100111111_{2}=1855. Determine, with proof, whether there exists a strictly increasing eight-term arithmetic progression a1,…,a8a_{1}, \ldots, a_{8} of odd positive integers such that r(a1),…,r(a8)r\left(a_{1}\right), \ldots, r\left(a_{8}\right) is an arithmetic progression in that order.
Plain-text mathematical notation (without MathML)
For any odd positive integer n, let r(n) be the odd positive integer such that the binary representation of r(n) is the binary representation of n written backwards. For example, r(2023)=r(11111100111₂)=11100111111₂=1855. Determine, with proof, whether there exists a strictly increasing eight-term arithmetic progression a₁,…,a₈ of odd positive integers such that r(a₁),…,r(a₈) is an arithmetic progression in that order.
Original LaTeX notation
For any odd positive integer $n$, let $r(n)$ be the odd positive integer such that the binary representation of $r(n)$ is the binary representation of $n$ written backwards. For example, $r(2023)=r\left(11111100111_{2}\right)=11100111111_{2}=1855$. Determine, with proof, whether there exists a strictly increasing eight-term arithmetic progression $a_{1}, \ldots, a_{8}$ of odd positive integers such that $r\left(a_{1}\right), \ldots, r\left(a_{8}\right)$ is an arithmetic progression in that order.

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
b4b9399d-1f56-5fb3-9f7c-ffb41c34fb32
task_key
test--b4b9399d-1f56-5fb3-9f7c-ffb41c34fb32
task_revision_id
1
config / split
/ test

Official source

initial import