MathDB

There is a central train station in point

O

, which is connected to other train stations

A_1, A_2, \ldots, A_8

with tracks. There is also a track between stations

A_i

and

A_{i+1}

for each

i

from

1

8

(here

A_9 = A_1

). The length of each track

A_iA_{i+1}

is equal to

1

, and the length of each track

OA_i

is equal to

2

, for each

i

from

1

8

.
There are also

8

trains

B_1, B_2, \ldots, B_8

, with speeds

1, 2, \ldots, 8

correspondently. Trains can move only by the tracks above, in both directions. No time is wasted on changing directions. If two or more trains meet at some point, they will move together from now on, with the speed equal to that of the fastest of them.
Is it possible to arrange trains into stations

A_1, A_2, \ldots, A_8

(each station has to contain one train initially), and to organize their movement in such a way, that all trains arrive at

O

in time

t < \frac{1}{2}

?
(Proposed by Bogdan Rublov)

Problem Statement