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OlympiadBench / 2045 / A circle ω of radius 1 is given. A collection T of triangles is called…

Problem

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

question

A circle

\omega

of radius 1 is given. A collection

T

of triangles is called good, if the following conditions hold: (i) each triangle from

T

is inscribed in

\omega

; (ii) no two triangles from

T

have a common interior point. Determine all positive real numbers

t

such that, for each positive integer

n

, there exists a good collection of

n

triangles, each of perimeter greater than

t

Plain-text mathematical notation (without MathML)

A circle ω of radius 1 is given. A collection T of triangles is called good, if the following conditions hold:

(i) each triangle from T is inscribed in ω;

(ii) no two triangles from T have a common interior point.

Determine all positive real numbers t such that, for each positive integer n, there exists a good collection of n triangles, each of perimeter greater than t.

Original LaTeX notation

A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:

(i) each triangle from $T$ is inscribed in $\omega$;

(ii) no two triangles from $T$ have a common interior point.

Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

answer type

Interval

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2045
task_id: b63c9ab6-2477-580d-89aa-c1dbb81a700b
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2045
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

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