MathDB

1

Part of 2002 IberoAmerican

Problems(2)

17th ibmo - el salvador 2002.

Source: Spanish Communities

4/14/2006
The integer numbers from 11 to 20022002 are written in a blackboard in increasing order 1,2,…,2001,20021,2,\ldots, 2001,2002. After that, somebody erases the numbers in the (3k+1)−th (3k+1)-th places i.e. (1,4,7,… )(1,4,7,\dots). After that, the same person erases the numbers in the (3k+1)−th(3k+1)-th positions of the new list (in this case, 2,5,9,…2,5,9,\ldots). This process is repeated until one number remains. What is this number?
number theory unsolvednumber theory
17th ibmo - el salvador 2002/q4.

Source: Spanish Communities

4/14/2006
In a triangle △ABC\triangle{ABC} with all its sides of different length, DD is on the side ACAC, such that BDBD is the angle bisector of ∢ABC\sphericalangle{ABC}. Let EE and FF, respectively, be the feet of the perpendicular drawn from AA and CC to the line BDBD and let MM be the point on BCBC such that DMDM is perpendicular to BCBC. Show that ∢EMD=∢DMF\sphericalangle{EMD}=\sphericalangle{DMF}.
geometryangle bisector