benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / There are n line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his 2n−1 friends each stand at a d…

Problem

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

problem

There are nn line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his 2n−12 n-1 friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a "sink". Then he places the present at the endpoint of the segment he is at. The present moves as follows: - If it is on a line segment, it moves towards the sink. - When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove Tony can send presents to exactly nn of his 2n−12 n-1 friends.
Plain-text mathematical notation (without MathML)
There are n line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his 2n−1 friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a "sink". Then he places the present at the endpoint of the segment he is at. The present moves as follows: - If it is on a line segment, it moves towards the sink. - When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove Tony can send presents to exactly n of his 2n−1 friends.
Original LaTeX notation
There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2 n-1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a "sink". Then he places the present at the endpoint of the segment he is at. The present moves as follows: - If it is on a line segment, it moves towards the sink. - When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove Tony can send presents to exactly $n$ of his $2 n-1$ friends.

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
55836edc-83f2-556d-9eed-7189ea3318aa
task_key
test--55836edc-83f2-556d-9eed-7189ea3318aa
task_revision_id
1
config / split
/ test

Official source

initial import