2

Part of 2006 Tuymaada Olympiad

Problems(2)

fibonacci sequences

Source: tuymaada 2006 - problem 2

We call a sequence of integers a Fibonacci-type sequence if it is infinite in both ways and

a_{n}=a_{n-1}+a_{n-2}

n\in\mathbb{Z}

. How many Fibonacci-type sequences can we find, with the property that in these sequences there are two consecutive terms, strictly positive, and less or equal than

N

? (two sequences are considered to be the same if they differ only by shifting of indices)
Proposed by I. Pevzner

inductionnumber theory unsolvednumber theory

an inequality in triangle

Source: tuymaada 2006 - problem 6

ABC

G

H

it`s orthocenter, and

M

the midpoint of the arc

\widehat{AC}

(not containing

B

). It is known that

MG=R

R

is the radius of the circumcircle. Prove that

BG\geq BH

.
Proposed by F. Bakharev

inequalitiesgeometrycircumcircleEulertrigonometrytriangle inequalitygeometry unsolved