Baltic Way 1996 · Problem 7

Number Theory

A sequence of integers $a_{1}, a_{2}, \ldots$ , is such that $a_{1}=1, a_{2}=2$ and for $n \geq 1$

a_{n+2}= \begin{cases}5 a_{n+1}-3 a_{n} & \text { if } a_{n} \cdot a_{n+1} \text { is even, } \\ a_{n+1}-a_{n} & \text { if } a_{n} \cdot a_{n+1} \text { is odd. }\end{cases}

Prove that $a_{n} \neq 0$ for all $n$ .

When you’re ready

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Review

Modular arithmetic

Solution

Solution: Considering the sequence modulo $6$ we obtain $1, 2, 1, 5, 4, 5, 1, 2, \ldots$ The conclusion follows.

Results from Baltic Way 1996

10 teams

Score distribution

All team scores