MathDB

Let

n

be a positive integer and let vectors

v_1

v_2

\ldots

v_n

be given in the plain. A flea originally sitting in the origin moves according to the following rule: in the

i

th minute (for

i=1,2,\ldots,n

) it will stay where it is with probability

1/2

, moves with vector

v_i

with probability

1/4

, and moves with vector

-v_i

with probability

1/4

. Prove that after the

n

th minute there exists no point which is occupied by the flea with greater probability than the origin.
Proposed by Péter Pál Pach, Budapest

Problem Statement