IMO Shortlist 2011 problem N8
Dodao/la:
arhiva23. lipnja 2013. Let
![k \in \mathbb{Z}^+](/media/m/d/1/2/d12e768b5574117f2bf390564cc4e3a3.png)
and set
![n=2^k+1.](/media/m/2/1/5/2154c214031283499a624ebf7d1bf1d1.png)
Prove that
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
is a prime number if and only if the following holds: there is a permutation
![a_{1},\ldots,a_{n-1}](/media/m/3/5/c/35c9f5c2f815f43f5037b2c03198279b.png)
of the numbers
![1,2, \ldots, n-1](/media/m/b/5/e/b5e64105024e067aa87d85a4e701ddce.png)
and a sequence of integers
![g_{1},\ldots,g_{n-1},](/media/m/a/a/0/aa0ba67d66cb77ea9e009aa7ed17ef66.png)
such that
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
divides
![g^{a_i}_i - a_{i+1}](/media/m/5/c/e/5ce2fe34691b17b2966340d57862495d.png)
for every
![i \in \{1,2,\ldots,n-1\},](/media/m/c/7/e/c7e78f8db7749b135978dcdfbf9cfa4a.png)
where we set
![a_n = a_1.](/media/m/9/4/7/9479f0f5534b556801c7a1a4619a7240.png)
Proposed by Vasily Astakhov, Russia
%V0
Let $k \in \mathbb{Z}^+$ and set $n=2^k+1.$ Prove that $n$ is a prime number if and only if the following holds: there is a permutation $a_{1},\ldots,a_{n-1}$ of the numbers $1,2, \ldots, n-1$ and a sequence of integers $g_{1},\ldots,g_{n-1},$ such that $n$ divides $g^{a_i}_i - a_{i+1}$ for every $i \in \{1,2,\ldots,n-1\},$ where we set $a_n = a_1.$
Proposed by Vasily Astakhov, Russia
Izvor: Međunarodna matematička olimpijada, shortlist 2011