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Can all terms of the sequence be distinct?

Source: Baltic Way 2005/1 - Proposed by ME :)

November 7, 2005
ceiling functionalgebra

Problem Statement

Let a0a_0 be a positive integer. Define the sequence {an}n≥0\{a_n\}_{n \geq 0} as follows: if an=∑i=0jci10i a_n = \sum_{i = 0}^jc_i10^i where ci∈{0,1,2,⋯ ,9}c_i \in \{0,1,2,\cdots,9\}, then an+1=c02005+c12005+⋯+cj2005. a_{n + 1} = c_0^{2005} + c_1^{2005} + \cdots + c_j^{2005}. Is it possible to choose a0a_0 such that all terms in the sequence are distinct?
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