MathDB

3

Part of 1998 Federal Competition For Advanced Students, Part 2

Problems(2)

Calculate the sum in the sequence

Source: Austrian Mathematical Olympiad 1998, Part 2, D1, P3

6/29/2011
Let ana_n be a sequence recursively de fined by a0=0,a1=1a_0 = 0, a_1 = 1 and an+2=an+1+ana_{n+2} = a_{n+1} + a_n. Calculate the sum of an(25)na_n\left( \frac 25\right)^n for all positive integers nn. For what value of the base bb we get the sum 11?
functioncombinatorics proposedcombinatorics
Relationship between the lengths of the diagonals AC and BD

Source: Austrian Mathematical Olympiad 1998, Part 2, D2, P3

6/29/2011
In a parallelogram ABCDABCD with the side ratio AB:BC=2:3AB : BC = 2 : \sqrt 3 the normal through DD to ACAC and the normal through CC to ABAB intersects in the point EE on the line ABAB. What is the relationship between the lengths of the diagonals ACAC and BDBD?
geometryparallelogramratiogeometry proposed