BMO Round 1, 2012

Combinatorics Difficulty 6.0 National Olympiad Prove it United Kingdom

Initially there are $m$ balls in one bag, and $n$ in the other, where $m, n > 0$ . Two different operations are allowed:

a) Remove an equal number of balls from each bag;

b) Double the number of balls in one bag.

Is it always possible to empty both bags after a finite sequence of operations?

Operation b) is now replaced with

b $'$ ) Triple the number of balls in one bag.

Is it now always possible to empty both bags after a finite sequence of operations?

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