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OlympiadBench / 2301 / For some positive integers k, the parabola with equation y=(x²)/(k)−5…

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

For some positive integers

k

, the parabola with equation

y=\frac{x^{2}}{k}-5

intersects the circle with equation

x^{2}+y^{2}=25

at exactly three distinct points

A, B

and

C

. Determine all such positive integers

k

for which the area of

\triangle A B C

is an integer.

Plain-text mathematical notation (without MathML)

For some positive integers k, the parabola with equation y=(x²)/(k)−5 intersects the circle with equation x²+y²=25 at exactly three distinct points A,B and C. Determine all such positive integers k for which the area of △ABC is an integer.

Original LaTeX notation

For some positive integers $k$, the parabola with equation $y=\frac{x^{2}}{k}-5$ intersects the circle with equation $x^{2}+y^{2}=25$ at exactly three distinct points $A, B$ and $C$. Determine all such positive integers $k$ for which the area of $\triangle A B C$ is an integer.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2301
task_id: 94beead7-844f-5e9b-bfc4-59ac72d70460
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2301
task_revision_id: 2
config / split: OE_TO_maths_en_COMP / train

Official source

initial import