1. The sum of the digits of a natural number is 2017. Moreover, any ten consecutive digits of the number are distinct. How many digits can the number have? List all possible answers and prove that there are no others.
Solution. Among every ten consecutive digits, all digits from 0 to 9 appear exactly once, meaning the sum of every 10 consecutive digits is 45.

Notice that $2017=44 \cdot 45+37$. This means the number consists of 44 blocks of 10 digits each, and at the end, there are a few more digits with a sum of 37. In other words, up to 45 blocks (450 digits), the sum of the digits is 8 short, which can be achieved by one (8), two (35), three (521), or four (5210) digits, but not more (since $0+1+2+3+4>8$).

Answer: from 446 to 449.