ISL 1985

Show that the sequence $\{a_n\}_{n\geq1}$ defined by $a_n = [n \sqrt 2]$ contains an infinite number of integer powers of $2$ . ( $[x]$ is the integer part of $x$ .)

Factorise $5^{1985}-1$ as a product of three integers, each greater than $5^{100}$ .