benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Consider a regular 2n-gon P , A₁,A₂,⋯,A_(2n) in the plane ,where n is a positive integer . We say that a point S on one of the sides of P can be seen from a point E that is externa…

Problem

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

problem

Consider a regular 2n-gon PP , A1,A2,⋯,A2nA_1,A_2,\cdots ,A_{2n} in the plane ,where nn is a positive integer . We say that a point SS on one of the sides of PP can be seen from a point EE that is external to PP , if the line segment SESE contains no other points that lie on the sides of PP except SS .We color the sides of PP in 3 different colors (ignore the vertices of PP ,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover ,from every point in the plane external to PP , points of most 2 different colors on PP can be seen .Find the number of distinct such colorings of PP (two colorings are considered distinct if at least one of sides is colored differently).
Plain-text mathematical notation (without MathML)
Consider a regular 2n-gon P , A₁,A₂,⋯,A_(2n) in the plane ,where n is a positive integer . We say that a point S on one of the sides of P can be seen from a point E that is external to P , if the line segment SE contains no other points that lie  on the sides of P except S .We color the sides of P in 3 different colors (ignore the vertices of P ,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover  ,from every point in the plane external to P , points of most 2  different colors on P can be seen .Find the number of distinct such colorings of P (two colorings are considered distinct if at least one of sides is colored differently).
Original LaTeX notation
Consider a regular 2n-gon $P$ , $A_1,A_2,\cdots ,A_{2n}$ in the plane ,where $n$ is a positive integer . We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $SE$ contains no other points that lie  on the sides of $P$ except $S$ .We color the sides of $P$ in 3 different colors (ignore the vertices of $P$ ,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover  ,from every point in the plane external to $P$ , points of most 2  different colors on $P$ can be seen .Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently).

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
9d67ea2e-c76e-5415-a153-4c60c368804f
task_key
test--9d67ea2e-c76e-5415-a153-4c60c368804f
task_revision_id
1
config / split
/ test

Official source

initial import