MathDB

2

Part of 2004 CentroAmerican

Problems(2)

Integers expressible as the sum of two terms in the sequence

Source: CentroAmerican 2004

12/10/2010
Define the sequence (an)(a_n) as follows: a0=a1=1a_0=a_1=1 and for k≥2k\ge 2, ak=ak−1+ak−2+1a_k=a_{k-1}+a_{k-2}+1. Determine how many integers between 11 and 20042004 inclusive can be expressed as am+ana_m+a_n with mm and nn positive integers and m≠nm\not= n.
algebra proposedalgebra
The point P in the trapezium ABCD

Source: CentroAmerican 2004

12/10/2010
Let ABCDABCD be a trapezium such that AB∣∣CDAB||CD and AB+CD=ADAB+CD=AD. Let PP be the point on ADAD such that AP=ABAP=AB and PD=CDPD=CD.
a)a) Prove that ∠BPC=90∘\angle BPC=90^{\circ}. b)b) QQ is the midpoint of BCBC and RR is the point of intersection between the line ADAD and the circle passing through the points B,AB,A and QQ. Show that the points B,P,RB,P,R and CC are concyclic.
geometrytrapezoidcircumcircleangle bisectorgeometry proposed