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2018 Canada National Olympiad
Problems
(1)
Collapsible arrangements [CMO 2018 - P1]
Source: 2018 Canadian Mathematical Olympiad - P1
3/31/2018
Consider an arrangement of tokens in the plane, not necessarily at distinct points. We are allowed to apply a sequence of moves of the following kind: select a pair of tokens at points
A
A
A
and
B
B
B
and move both of them to the midpoint of
A
A
A
and
B
B
B
. We say that an arrangement of
n
n
n
tokens is collapsible if it is possible to end up with all
n
n
n
tokens at the same point after a finite number of moves. Prove that every arrangement of
n
n
n
tokens is collapsible if and only if
n
n
n
is a power of
2
2
2
.
combinatorics