IMO Shortlist 1999 problem N4

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Dodao/la: arhiva
2. travnja 2012.
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
Izvor: Međunarodna matematička olimpijada, shortlist 1999