Question 102
Let $a_{0}$ be a positive integer. Define the sequence $a_{n}$, $n \geq 0$, as follows: If
$$
a_{n} = \sum_{i=0}^{j} c_{i} 10^{i}
$$
where $c_{i}$ are integers with $0 \leq c_{i} \leq 9$, then
$$
a_{n+1} = c_{0}^{2005} + c_{1}^{2005} + \cdots + c_{j}^{2005}.
$$
Is it possible to choose $a_{0}$ so that all the terms in the sequence are distinct?
$$
a_{n} = \sum_{i=0}^{j} c_{i} 10^{i}
$$
where $c_{i}$ are integers with $0 \leq c_{i} \leq 9$, then
$$
a_{n+1} = c_{0}^{2005} + c_{1}^{2005} + \cdots + c_{j}^{2005}.
$$
Is it possible to choose $a_{0}$ so that all the terms in the sequence are distinct?