RU-ALLRUSSIAN 1976 Problem 10

Initially, there is one beetle on each square in the set S. Suddenly each beetle flies to a new square, subject to the following conditions: (1) the new square may be the same as the old or different; (2) more than one beetle may choose the same new square; (3) if two beetles are initially in squares with a common vertex, then after the flight they are either in the same square or in squares with a common vertex. Suppose S is the set of all squares in the middle row and column of a $99 \times 99$ chess board, is it true that there must always be a beetle whose new square shares a vertex with its old square (or is identical with it)? What if S also includes all the border squares (so S is rows 1, 50 and 99 and columns 1, 50 and 99)? What if S is all squares of the board?