MathDB

2

Part of 1992 Turkey Team Selection Test

Problems(2)

Turkey TST 1992 - P5: m balls n boxes

Source:

3/13/2011
There are nn boxes which is numbere from 11 to nn. The box with number 11 is open, and the others are closed. There are mm identical balls (m≥nm\geq n). One of the balls is put into the open box, then we open the box with number 22. Now, we put another ball to one of two open boxes, then we open the box with number 33. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes. In how many ways this arrangement can be done?
inductioninequalitiescombinatorics proposedcombinatorics
Turkey TST 1992 - P2: Altitude meets circumcircle

Source:

3/13/2011
The line passing through BB is perpendicular to the side ACAC at EE. This line meets the circumcircle of △ABC\triangle ABC at DD. The foot of the perpendicular from DD to the side BCBC is FF. If OO is the center of the circumcircle of △ABC\triangle ABC, prove that BOBO is perpendicular to EFEF.
geometrycircumcirclegeometry proposed