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Problems
Contests
International Contests
Tournament Of Towns
1995 Tournament Of Towns
(442) 2
(442) 2
Part of
1995 Tournament Of Towns
Problems
(1)
three grasshoppers on a line
Source: TOT 442 1995 Spring O J2 - Tournament of Towns
6/12/2024
Three grasshoppers
A
A
A
,
B
B
B
and
C
C
C
are placed on a line. Grasshopper
B
B
B
sits at the midpoint between
A
A
A
and
C
C
C
. Every second, one of the grasshoppers jumps over one of the others to the symmetrical point on the other side (if
X
X
X
jumps over
Y
Y
Y
to point
X
′
X'
X
′
, then
X
Y
XY
X
Y
=
Y
X
′
YX'
Y
X
′
). After several jumps it so happened that they returned to the three initial points (but maybe in different order). Prove that in this case
B
B
B
returns to his initial middle position.(AK Kovaldzhy)
combinatorics