MathDB

2

Part of 2010 Mexico National Olympiad

Problems(2)

Lights on a Board

Source: OMM 2010 2

7/15/2014
In each cell of an n×nn\times n board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off).
Show that if after a certain number of moves, at least one light is on, then at this moment at least nn lights are on.
analytic geometryinvariantinductioncombinatorics unsolvedcombinatorics
Right Angle in Acute Triangle...?

Source: OMM 2010 5

7/15/2014
Let ABCABC be an acute triangle with AB≠ACAB\neq AC, MM be the median of BCBC, and HH be the orthocenter of △ABC\triangle ABC. The circumcircle of BB, HH, and CC intersects the median AMAM at NN. Show that ∠ANH=90∘\angle ANH=90^\circ.
geometrycircumcirclepower of a pointradical axisgeometry unsolved