MathDB

2

Part of 2008 Pan African

Problems(2)

PA^2+PB^2=4PT^2

Source: Pan African Olympiad 2008

10/1/2011
Let C1C_1 be a circle with centre OO, and let ABAB be a chord of the circle that is not a diameter. MM is the midpoint of ABAB. Consider a point TT on the circle C2C_2 with diameter OMOM. The tangent to C2C_2 at the point TT intersects C1C_1 at two points. Let PP be one of these points. Show that PA2+PB2=4PT2PA^2+PB^2=4PT^2.
geometry proposedgeometry
Number of connected subsets of {1,2, ... ,10}

Source: Pan African Olympiad 2008

10/1/2011
A set of positive integers XX is called connected if ∣X∣≥2|X|\ge 2 and there exist two distinct elements mm and nn of XX such that mm is a divisor of nn. Determine the number of connected subsets of the set {1,2,…,10}\{1,2,\ldots,10\}.
combinatorics proposedcombinatorics