IMO Shortlist 2001 problem N4
Dodao/la:
arhiva2. travnja 2012. Let
![p \geq 5](/media/m/f/3/e/f3e75cfd3e2db080098f3fa1c97e064c.png)
be a prime number. Prove that there exists an integer
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
with
![1 \leq a \leq p-2](/media/m/5/4/0/5408263aba49b9d928d9f289b85c590e.png)
such that neither
![a^{p-1}-1](/media/m/1/7/b/17b7a505025eb40b518fcf753d15f043.png)
nor
![(a+1)^{p-1}-1](/media/m/f/2/1/f2136e80a7d14b6a8c51f4b2a7bd32d0.png)
is divisible by
![p^2](/media/m/f/0/f/f0fbfedc204c557f55c06eceeb024b6c.png)
.
%V0
Let $p \geq 5$ be a prime number. Prove that there exists an integer $a$ with $1 \leq a \leq p-2$ such that neither $a^{p-1}-1$ nor $(a+1)^{p-1}-1$ is divisible by $p^2$.
Izvor: Međunarodna matematička olimpijada, shortlist 2001