MathDB

1

Part of 2009 IMO Shortlist

Problems(2)

IMO Shortlist 2009 - Problem A1

Source:

7/5/2010
Find the largest possible integer kk, such that the following statement is true: Let 20092009 arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain b1≤b2≤…≤b2009the lengths of the blue sides r1≤r2≤…≤r2009the lengths of the red sides and w1≤w2≤…≤w2009the lengths of the white sides  \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right. Then there exist kk indices jj such that we can form a non-degenerated triangle with side lengths bjb_j, rjr_j, wjw_j.
Proposed by Michal Rolinek, Czech Republic
algebraIMO Shortlisttriangle inequality
IMO Shortlist 2009 - Problem C1

Source:

7/5/2010
Consider 20092009 cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of 5050 consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player?
Proposed by Michael Albert, Richard Guy, New Zealand
combinatoricsgameIMO Shortlistgame strategyConceptual