4

Part of 2002 Irish Math Olympiad

Problems(2)

positive integer

Source: Ireland 2002

(a_n)

is defined by a_1\equal{}a_2\equal{}a_3\equal{}1 and a_{n\plus{}1}a_{n\minus{}2}\minus{}a_n a_{n\minus{}1}\equal{}2 for all

n \ge 3.

a_n

is a positive integer for all

n \ge 1

number theory unsolvednumber theory

Source: Ireland 2002

Let \alpha\equal{}2\plus{}\sqrt{3}. Prove that \alpha^n\minus{}[\alpha^n]\equal{}1\minus{}\alpha^{\minus{}n} for all

n \in \mathbb{N}_0

floor functionalgebra unsolvedalgebra