benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Find all positive integers n such that the following statement holds: Suppose real numbers a₁, a₂, …, a_(n), b₁, b₂, …, b_(n) satisfy |a_(k)|+|b_(k)|=1 for all k=1,…,n. Then there …

Problem

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

problem

Find all positive integers nn such that the following statement holds: Suppose real numbers a1a_1, a2a_2, …\dots, ana_n, b1b_1, b2b_2, …\dots, bnb_n satisfy |ak|+|bk|=1|a_k|+|b_k|=1 for all k=1,…,nk=1,\dots,n. Then there exists ε1\varepsilon_1, ε2\varepsilon_2, …\dots, εn\varepsilon_n, each of which is either −1-1 or 11, such that |∑i=1nεiai|+|∑i=1nεibi|≤1. \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1.
Plain-text mathematical notation (without MathML)
Find all positive integers n such that the following statement holds: Suppose real numbers a₁, a₂, …, a_(n), b₁, b₂, …, b_(n) satisfy |a_(k)|+|b_(k)|=1 for all k=1,…,n. Then there exists ε₁, ε₂, …, ε_(n), each of which is either −1 or 1, such that
    |∑_(i=1)^(n)ε_(i)a_(i)|+|∑_(i=1)^(n)ε_(i)b_(i)|≤1.
Original LaTeX notation
Find all positive integers $n$ such that the following statement holds: Suppose real numbers $a_1$, $a_2$, $\dots$, $a_n$, $b_1$, $b_2$, $\dots$, $b_n$ satisfy $|a_k|+|b_k|=1$ for all $k=1,\dots,n$. Then there exists $\varepsilon_1$, $\varepsilon_2$, $\dots$, $\varepsilon_n$, each of which is either $-1$ or $1$, such that
    \[ \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1. \]

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
85f32d9e-31f9-5ae0-8ef9-1aab826c22b4
task_key
test--85f32d9e-31f9-5ae0-8ef9-1aab826c22b4
task_revision_id
1
config / split
/ test

Official source

initial import