IMO Shortlist 1999 problem N4
Dodao/la:
arhiva2. travnja 2012. Denote by S the set of all primes such the decimal representation of
![\frac{1}{p}](/media/m/b/5/a/b5ac5bed9271c6557d7b4ed8cc2cbf2e.png)
has the fundamental period divisible by 3. For every
![p \in S](/media/m/2/e/9/2e9f55a1f78b5a93f8eae084c0c173cd.png)
such that
![\frac{1}{p}](/media/m/b/5/a/b5ac5bed9271c6557d7b4ed8cc2cbf2e.png)
has the fundamental period
![3r](/media/m/a/4/1/a411f7a7965bc8fd6dc1f341ed5de729.png)
one may write
where
![r=r(p)](/media/m/f/c/1/fc109f56251be2280a5382cacb4ae4b7.png)
; for every
![p \in S](/media/m/2/e/9/2e9f55a1f78b5a93f8eae084c0c173cd.png)
and every integer
![k \geq 1](/media/m/4/7/4/474e08320ac4dfd51b6214797b6d06be.png)
define
![f(k,p)](/media/m/5/a/a/5aa8a885cf403ed00683c726f9587fee.png)
by
a) Prove that
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
is infinite.
b) Find the highest value of
![f(k,p)](/media/m/5/a/a/5aa8a885cf403ed00683c726f9587fee.png)
for
![k \geq 1](/media/m/4/7/4/474e08320ac4dfd51b6214797b6d06be.png)
and
%V0
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