MathDB

3

Part of 2009 Pan African

Problems(2)

x must be an integer

Source: Pan African Olympiad 2009

10/1/2011
Let xx be a real number with the following property: for each positive integer qq, there exists an integer pp, such that ∣x−pq∣<13q.\left|x-\frac{p}{q} \right|<\frac{1}{3q}. Prove that xx is an integer.
number theory proposednumber theory
Geometry Lots of Tangents and Chords

Source:

4/25/2011
Points C,E,DC,E,D and FF lie on a circle with centre OO. Two chords CDCD and EFEF intersect at a point NN. The tangents at CC and DD intersect at AA, and the tangents at EE and FF intersect at BB. Prove that ON⊥ABON\perp AB.
geometry