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OlympiadBench / 1807 / Let q be a real number. Gugu has a napkin with ten distinct real numbers…

Problem

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

answer type

Numerical

is multiple answer

true

language

English

question

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: - 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

q a b

, 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. 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.

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:

- 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.

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:

- 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 $q a b$, 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.

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.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1807
task_id: 4b4711a1-6e3c-5260-8f86-3232d1bb312b
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1807
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

initial import