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28

Part of 2012 National Olympiad First Round

Problems(1)

Turkish NMO First Round - 2012 Problem - 28 {Combinatorics}

Source:

7/1/2012
At the beginning, three boxes contain mm, nn, and kk pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing the game once for each (m,n,k)=(1,2012,2014)(m,n,k)=(1,2012,2014), (2011,2011,2012)(2011,2011,2012), (2011,2012,2013)(2011,2012,2013), (2011,2012,2014)(2011,2012,2014), (2011,2013,2013)(2011,2013,2013). In how many of them can Ayşe guarantee to win the game?
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 4<spanclass=′latex−bold′>(E)</span> 5 <span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ 4 \qquad <span class='latex-bold'>(E)</span>\ 5