benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / On a party with 99 guests, hosts Ann and Bob play a game (the hosts are not…

Problem

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

problem

On a party with 99 guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are 99 chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair cc. If some chair adjacent to cc is already occupied, the same host orders one guest on such chair to stand up (if both chairs adjacent to cc are occupied, the host chooses exactly one of them). All orders are carried out immediately. Ann makes the first move; her goal is to fulfill, after some move of hers, that at least kk chairs are occupied. Determine the largest kk for which Ann can reach the goal, regardless of Bob's play.
Plain-text mathematical notation (without MathML)
On a party with 99 guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are 99 chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair c. If some chair adjacent to c is already occupied, the same host orders one guest on such chair to stand up (if both chairs adjacent to c are occupied, the host chooses exactly one of them). All orders are carried out immediately. Ann makes the first move; her goal is to fulfill, after some move of hers, that at least k chairs are occupied. Determine the largest k for which Ann can reach the goal, regardless of Bob's play.
Original LaTeX notation
On a party with 99 guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are 99 chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair $c$. If some chair adjacent to $c$ is already occupied, the same host orders one guest on such chair to stand up (if both chairs adjacent to $c$ are occupied, the host chooses exactly one of them). All orders are carried out immediately. Ann makes the first move; her goal is to fulfill, after some move of hers, that at least $k$ chairs are occupied. Determine the largest $k$ for which Ann can reach the goal, regardless of Bob's play.

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
f596f949-824e-5c98-a947-fd24ac84527e
task_key
test--f596f949-824e-5c98-a947-fd24ac84527e
task_revision_id
1
config / split
/ test

Official source

initial import