benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / For a positive integer n, let θ(n) denote the number of integers 0≤x<2010 such that x²−n is divisible by 2010. Determine the remainder when ∑_(n=0)²⁰⁰⁹n⋅θ(n) is divided by 2010.

Problem

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

problem

For a positive integer nn, let θ(n)\theta(n) denote the number of integers 0≤x<20100 \leq x<2010 such that x2−nx^{2}-n is divisible by 2010. Determine the remainder when ∑n=02009n⋅θ(n)\sum_{n=0}^{2009} n \cdot \theta(n) is divided by 2010.
Plain-text mathematical notation (without MathML)
For a positive integer n, let θ(n) denote the number of integers 0≤x<2010 such that x²−n is divisible by 2010. Determine the remainder when ∑_(n=0)²⁰⁰⁹n⋅θ(n) is divided by 2010.
Original LaTeX notation
For a positive integer $n$, let $\theta(n)$ denote the number of integers $0 \leq x<2010$ such that $x^{2}-n$ is divisible by 2010. Determine the remainder when $\sum_{n=0}^{2009} n \cdot \theta(n)$ is divided by 2010.

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
94489021-7935-53d7-a8de-f268f4af4a79
task_key
test--94489021-7935-53d7-a8de-f268f4af4a79
task_revision_id
1
config / split
/ test

Official source

initial import