MathDB

1

Part of 2009 Bosnia Herzegovina Team Selection Test

Problems(2)

Bosnia 2009 Problem 1

Source:

8/10/2010
Denote by MM and NN feets of perpendiculars from AA to angle bisectors of exterior angles at BB and C,C, in triangle △ABC.\triangle ABC. Prove that the length of segment MNMN is equal to semiperimeter of triangle △ABC.\triangle ABC.
ratioexterior anglegeometry proposedgeometry
1xn table

Source: Bosnia 2009., problem 4.

8/6/2011
Given an 11 x nn table (n≥2n\geq 2), two players alternate the moves in which they write the signs + and - in the cells of the table. The first player always writes +, while the second always writes -. It is not allowed for two equal signs to appear in the adjacent cells. The player who can’t make a move loses the game. Which of the players has a winning strategy?
inductioncombinatorics unsolvedcombinatoricsGame TheoryBosnia2009