IMO 2010 Problem 5

Each of the six boxes $B_1$ , $B_2$ , $B_3$ , $B_4$ , $B_5$ , $B_6$ initially contains one coin. The following operations are allowed

Type 1) Choose a non-empty box $B_j$ , $1\leq j \leq 5$ , remove one coin from $B_j$ and add two coins to $B_{j+1}$ ;

Type 2) Choose a non-empty box $B_k$ , $1\leq k \leq 4$ , remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$ .

Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$ , $B_2$ , $B_3$ , $B_4$ , $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.

Proposed by Hans Zantema, Netherlands