IMO Shortlist 2002 problem A5
Dodao/la:
arhiva2. travnja 2012. Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer that is not a perfect cube. Define real numbers
![a,b,c](/media/m/3/6/4/36454fdb50fc50f021324b33a6b513e3.png)
by
where
![[x]](/media/m/6/a/4/6a47dfb91475b9d5490dbb3a666604a3.png)
denotes the integer part of
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
. Prove that there are infinitely many such integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
with the property that there exist integers
![r,s,t](/media/m/8/4/2/842bdc4f0a0f5726c647eb860480a9f6.png)
, not all zero, such that
![ra+sb+tc=0](/media/m/e/e/d/eed5030a126b5bb8264adaabf89d48d6.png)
.
%V0
Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by
$$a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,$$
where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.
Izvor: Međunarodna matematička olimpijada, shortlist 2002