MathDB

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Part of 2000 Federal Competition For Advanced Students, Part 2

Problems(2)

Prove that the Euler line HU bisects the angle BHD

Source: Austrian Mathematical Olympiad 2000, Part 2, D2, P1

6/28/2011
In a non-equilateral acute-angled triangle ABCABC with ∠C=60∘\angle C = 60^\circ, UU is the circumcenter, HH the orthocenter and DD the intersection of AHAH and BCBC. Prove that the Euler line HUHU bisects the angle BHDBHD.
Eulergeometrycircumcircleangle bisectorgeometry proposed
Find the explicit form of the sequence b_n

Source: Austrian Mathematical Olympiad 2000, Part 2, D1, P1

6/28/2011
The sequence an is defined by a0=4,a1=1a_0 = 4, a_1 = 1 and the recurrence formula an+1=an+6an−1a_{n+1} = a_n + 6a_{n-1}. The sequence bnb_n is given by bn=∑k=0n(nk)ak.b_n=\sum_{k=0}^n \binom nk a_k. Find the coefficients α,β\alpha,\beta so that bnb_n satisfies the recurrence formula b_{n+1} = \alpha b_n + \beta b_{n-1}. Find the explicit form of bnb_n.
algebrabinomial theoremalgebra proposed