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2010 IMO Problems/Problem 5

Revision as of 10:13, 20 July 2010 by Bugi (talk | contribs) (Created page with '== Problem == Each of the six boxes <math>B_1</math>, <math>B_2</math>, <math>B_3</math>, <math>B_4</math>, <math>B_5</math>, <math>B_6</math> initially contains one coin. The f…')
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Problem

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.

Author: Unknown currently

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