benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / 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, re…

Problem

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

problem

Determine the smallest positive real number k k with the following property. Let ABCD ABCD be a convex quadrilateral, and let points A1 A_1, B1 B_1, C1 C_1, and D1 D_1 lie on sides AB AB, BC BC, CD CD, and DA DA, respectively. Consider the areas of triangles AA1D1 AA_1D_1, BB1A1 BB_1A_1, CC1B1 CC_1B_1 and DD1C1 DD_1C_1; let S S be the sum of the two smallest ones, and let S1 S_1 be the area of quadrilateral A1B1C1D1 A_1B_1C_1D_1. Then we always have kS1≥S kS_1\ge S. [i]Author: Zuming Feng and Oleg Golberg, USA[/i]
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.

[i]Author: Zuming Feng and Oleg Golberg, USA[/i]
Original LaTeX notation
Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_1$, $ CC_1B_1$ and $ DD_1C_1$; let $ S$ be the sum of the two smallest ones, and let $ S_1$ be the area of quadrilateral $ A_1B_1C_1D_1$. Then we always have $ kS_1\ge S$.

[i]Author: Zuming Feng and Oleg Golberg, USA[/i]

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
af36bc7c-3bfa-5db1-9751-094cacb22f77
task_key
test--af36bc7c-3bfa-5db1-9751-094cacb22f77
task_revision_id
1
config / split
/ test

Official source

initial import