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OlympiadBench / 1675 / Determine the smallest positive real number k with the following property.

Problem

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

answer type

Numerical

is multiple answer

false

language

English

question

Determine the smallest positive real number

k

with the following property. Let

A B C D

be a convex quadrilateral, and let points

A_{1}, B_{1}, C_{1}

and

D_{1}

lie on sides

A B, B C

C D

and

D A

, respectively. Consider the areas of triangles

A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B_{1}

, and

D D_{1} C_{1}

; let

S

be the sum of the two smallest ones, and let

S_{1}

be the area of quadrilateral

A_{1} B_{1} C_{1} D_{1}

. Then we always have

k S_{1} \geq S

Plain-text mathematical notation (without MathML)

Determine the smallest positive real number k with the following property.

Let ABCD be a convex quadrilateral, and let points A₁,B₁,C₁ and D₁ lie on sides AB,BC, CD and DA, respectively. Consider the areas of triangles AA₁D₁,BB₁A₁,CC₁B₁, and DD₁C₁; let S be the sum of the two smallest ones, and let S₁ be the area of quadrilateral A₁B₁C₁D₁. Then we always have kS₁≥S.

Original LaTeX notation

Determine the smallest positive real number $k$ with the following property.

Let $A B C D$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $A B, B C$, $C D$ and $D A$, respectively. Consider the areas of triangles $A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B_{1}$, and $D D_{1} C_{1}$; let $S$ be the sum of the two smallest ones, and let $S_{1}$ be the area of quadrilateral $A_{1} B_{1} C_{1} D_{1}$. Then we always have $k S_{1} \geq S$.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1675
task_id: 6828c457-a6ad-576d-bec6-0d6279a8504a
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1675
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

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