MathDB

2

Part of 2009 Junior Balkan Team Selection Test

Problems(2)

Ratio in isosceles right triangle

Source: Serbian JBTST 2, Day 1

5/30/2009
In isosceles right triangle ABC ABC a circle is inscribed. Let CD CD be the hypotenuse height (D∈AB D\in AB), and let P P be the intersection of inscribed circle and height CD CD. In which ratio does the circle divide segment AP AP?
ratiotrigonometrygeometry proposedgeometry
Choosing numbers with condition

Source: Serbian JBTST 2, Day 2

5/31/2009
From the set {1,2,3,…,2009} \{1,2,3,\ldots,2009\} we choose 1005 1005 numbers, such that sum of any 2 2 numbers isn't neither 2009 2009 nor 2010 2010. Find all ways on we can choose these 1005 1005 numbers.
combinatorics proposedcombinatorics