benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Let q be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the …

Problem

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

problem

Let qq be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the first line, Gugu writes down every number of the form a−ba-b, where aa and bb are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form qabqab, where aa and bb are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form a2+b2−c2−d2a^2+b^2-c^2-d^2, where a,b,c,da, b, c, d are four (not necessarily distinct) numbers from the first line. [/list] Determine all values of qq such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.
Plain-text mathematical notation (without MathML)
Let q be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
[list]
[*]In the first line, Gugu writes down every number of the form a−b, where a and b are two (not necessarily distinct) numbers on his napkin.
[*]In the second line, Gugu writes down every number of the form qab, where a and b are
two (not necessarily distinct) numbers from the first line.
[*]In the third line, Gugu writes down every number of the form a²+b²−c²−d², where a,b,c,d are four (not necessarily distinct) numbers from the first line.
[/list]
Determine all values of q such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.
Original LaTeX notation
Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
[list]
[*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
[*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are
two (not necessarily distinct) numbers from the first line.
[*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.
[/list]
Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

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
d1ba24ef-3aad-5701-aa97-fec67cf1e312
task_key
test--d1ba24ef-3aad-5701-aa97-fec67cf1e312
task_revision_id
1
config / split
/ test

Official source

initial import