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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.

question

Determine the smallest positive real number kk with the following property. Let ABCDA B C D be a convex quadrilateral, and let points A1,B1,C1A_{1}, B_{1}, C_{1} and D1D_{1} lie on sides AB,BCA B, B C, CDC D and DAD A, respectively. Consider the areas of triangles AA1D1,BB1A1,CC1B1A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B_{1}, and DD1C1D D_{1} C_{1}; let SS be the sum of the two smallest ones, and let S1S_{1} be the area of quadrilateral A1B1C1D1A_{1} B_{1} C_{1} D_{1}. Then we always have kS1≥Sk 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 $k S_{1} \geq S$.

answer type

Numerical

is multiple answer

false

language

English

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
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import