MathDB

3

Part of 2002 IberoAmerican

Problems(2)

17th ibmo - el salvador 2002/q3

Source: Spanish Communities

4/14/2006
Let PP be a point in the interior of the equilateral triangle △ABC\triangle ABC such that ∢APC=120∘\sphericalangle{APC}=120^\circ. Let MM be the intersection of CPCP with ABAB, and NN the intersection of APAP and BCBC. Find the locus of the circumcentre of the triangle MBNMBN as PP varies.
geometrycircumcirclesymmetryperpendicular bisectorgeometry unsolved
17th ibmo - el salvador 2002/q6.

Source: Spanish Communities

4/14/2006
A policeman is trying to catch a robber on a board of 2001×20012001\times2001 squares. They play alternately, and the player whose trun it is moves to a space in one of the following directions: ↓,→,↖\downarrow,\rightarrow,\nwarrow.
If the policeman is on the square in the bottom-right corner, he can go directly to the square in the upper-left corner (the robber can not do this). Initially the policeman is in the central square, and the robber is in the upper-left adjacent square. Show that:
a)a) The robber may move at least 1000010000 times before the being captured. b)b) The policeman has an strategy such that he will eventually catch the robber.
Note: The policeman can catch the robber if he reaches the square where the robber is, but not if the robber enters the square occupied by the policeman.
combinatorics unsolvedcombinatorics