IMO Shortlist 1988 problem 7
Dodao/la:
arhiva2. travnja 2012. Let
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
be the greatest positive root of the equation
![x^3 - 3 \cdot x^2 + 1 = 0.](/media/m/1/7/c/17c3682c57874ac07f87f7317ba972d0.png)
Show that
![\left[a^{1788} \right]](/media/m/2/0/b/20bbab2fd511dee40dec37fc2c138654.png)
and
![\left[a^{1988} \right]](/media/m/8/3/a/83a03eaaa1a03ce77e26e1535bddebfb.png)
are both divisible by 17. Here
![[x]](/media/m/6/a/4/6a47dfb91475b9d5490dbb3a666604a3.png)
denotes the integer part of
%V0
Let $a$ be the greatest positive root of the equation $x^3 - 3 \cdot x^2 + 1 = 0.$ Show that $\left[a^{1788} \right]$ and $\left[a^{1988} \right]$ are both divisible by 17. Here $[x]$ denotes the integer part of $x.$
Izvor: Međunarodna matematička olimpijada, shortlist 1988