ISL 1999 N4

Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write

$$\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots ,$$

where $r=r(p)$ ; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by $$f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}$$

a) Prove that $S$ is infinite. b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$ Attachments: