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OlympiadBench / 1695 / Let m be a positive integer and consider a checkerboard consisting of m by…

Problem

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

answer type

Expression

is multiple answer

false

language

English

question

Let

m

be a positive integer and consider a checkerboard consisting of

m

m

unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn

90^{\circ}

clockwise and continue moving with speed 1 . When more than two ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear. Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard or prove that such a moment does not necessarily exist.

Plain-text mathematical notation (without MathML)

Let m be a positive integer and consider a checkerboard consisting of m by m unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn 90^(∘) clockwise and continue moving with speed 1 . When more than two ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.

Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard or prove that such a moment does not necessarily exist.

Original LaTeX notation

Let $m$ be a positive integer and consider a checkerboard consisting of $m$ by $m$ unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed 1 . When more than two ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.

Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard or prove that such a moment does not necessarily exist.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1695
task_id: 6897df70-a9cf-5fe3-940d-f6249c4ab0dd
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1695
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

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