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Omni-MATH / A given finite number of lines in the plane, no two of which are parallel and no…

Problem

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

problem

A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write 11 or −1-1. In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines in the collection) and multiply by −1-1 each of the numbers in the triangle. Determine if it is always possible to obtain 11 in all the finite regions by successively applying this operation, regardless of the initial distribution of 11s and −1-1s.
Plain-text mathematical notation (without MathML)
A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write 1 or −1. In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines in the collection) and multiply by −1 each of the numbers in the triangle. Determine if it is always possible to obtain 1 in all the finite regions by successively applying this operation, regardless of the initial distribution of  1s and −1s.
Original LaTeX notation
A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write $1$ or $-1$. In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines in the collection) and multiply by $-1$ each of the numbers in the triangle. Determine if it is always possible to obtain $1$ in all the finite regions by successively applying this operation, regardless of the initial distribution of  $1$s and $-1$s.

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Source and history

task_id
36ec4ce4-b582-5170-ab8d-6bdf34dcc3b5
task_key
test--36ec4ce4-b582-5170-ab8d-6bdf34dcc3b5
task_revision_id
1
config / split
/ test

Official source

initial import