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Iran MO (2nd Round)
2021 Iran MO (2nd Round)
1
1
Part of
2021 Iran MO (2nd Round)
Problems
(1)
Two Points on a Line , can we reach the midpoint?
Source: 2021 Iran second round mathematical Olympiad P1
5/15/2021
There are two distinct Points
A
A
A
and
B
B
B
on a line. We color a point
P
P
P
on segment
A
B
AB
A
B
, distinct from
A
,
B
A,B
A
,
B
and midpoint of segment
A
B
AB
A
B
to red. In each move , we can reflect one of the red point wrt
A
A
A
or
B
B
B
and color the midpoint of the resulting point and the point we reflected from ( which is one of
A
A
A
or
B
B
B
) to red. For example , if we choose
P
P
P
and the reflection of
P
P
P
wrt to
A
A
A
is
P
′
P'
P
′
, then midpoint of
A
P
′
AP'
A
P
′
would be red. Is it possible to make the midpoint of
A
B
AB
A
B
red after a finite number of moves?
combinatorics
geometry