IMO Shortlist 2003 problem N4
Dodao/la:
arhiva2. travnja 2012. Let
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
be an integer greater than
![5](/media/m/e/a/3/ea36c795dac330f34d395d8364d379b6.png)
. For each positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
, consider the number
written in base
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
.
Prove that the following condition holds if and only if
![b = 10](/media/m/8/4/3/8438a2ab8509868e6107f731137e59b2.png)
: there exists a positive integer
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
such that for any integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
greater than
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
, the number
![x_n](/media/m/3/c/5/3c57e4750d576aafa08c9ec1a939cfce.png)
is a perfect square.
%V0
Let $b$ be an integer greater than $5$. For each positive integer $n$, consider the number
$$x_n = \underbrace{11\cdots1}_{n - 1}\underbrace{22\cdots2}_{n}5,$$
written in base $b$.
Prove that the following condition holds if and only if $b = 10$: there exists a positive integer $M$ such that for any integer $n$ greater than $M$, the number $x_n$ is a perfect square.
Izvor: Međunarodna matematička olimpijada, shortlist 2003