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OlympiadBench / 2198 / The n contestants of an EGMO are named C₁,…,C_(n). After the…

Problem

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

question

The

n

contestants of an EGMO are named

C_{1}, \ldots, C_{n}

. After the competition they queue in front of the restaurant according to the following rules. - The Jury chooses the initial order of the contestants in the queue. - Every minute, the Jury chooses an integer

i

with

1 \leq i \leq n

. - If contestant

C_{i}

has at least

i

other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly

i

positions. - If contestant

C_{i}

has fewer than

i

other contestants in front of her, the restaurant opens and the process ends. Determine for every

n

the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.

Plain-text mathematical notation (without MathML)

The n contestants of an EGMO are named C₁,…,C_(n). After the competition they queue in front of the restaurant according to the following rules.

- The Jury chooses the initial order of the contestants in the queue.
- Every minute, the Jury chooses an integer i with 1≤i≤n.
    - If contestant C_(i) has at least i other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly i positions.
    - If contestant C_(i) has fewer than i other contestants in front of her, the restaurant opens and the process ends.
Determine for every n the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.

Original LaTeX notation

The $n$ contestants of an EGMO are named $C_{1}, \ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules.

- The Jury chooses the initial order of the contestants in the queue.
- Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$.
    - If contestant $C_{i}$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions.
    - If contestant $C_{i}$ has fewer than $i$ other contestants in front of her, the restaurant opens and the process ends.
Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2198
task_id: ebd11d23-4ce7-530d-8608-5f2b30e3d992
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2198
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

initial import