MathDB

1

Part of 1996 China National Olympiad

Problems(2)

Orthocentre is collinear with two tangent points

Source: Chinese MO 1996

7/29/2010
Let △ABC\triangle{ABC} be a triangle with orthocentre HH. The tangent lines from AA to the circle with diameter BCBC touch this circle at PP and QQ. Prove that H,PH,P and QQ are collinear.
geometrycircumcircletrigonometryconicssimilar trianglesChina
Singers and concerts

Source: CMO-1996-4

1/3/2006
88 singers take part in a festival. The organiser wants to plan mm concerts. For every concert there are 44 singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises mm.
combinatorics unsolvedcombinatorics