Home > Algebra > IMC 2014 Day 2 Problem 1

## IMC 2014 Day 2 Problem 1

For a positive integer ${x}$ denote its ${n}$-th decimal digit by ${d_n(x)}$, i.e. ${d_n(x) \in \{0,1,..,9\}}$ and ${x = \displaystyle \sum_{n=1}^\infty d_n(x) 10^{n-1}}$. Suppose that for some sequence ${(a_n)_{n=1}^\infty}$ there are only finitely many zeros in the sequence ${(d_n(a_n))_{n=1}^\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence ${(a_n)_{n=1}^\infty}$.

IMC 2014 Day 2 Problem 1

Hint: Since $d_n(a_n)\neq 0$ it follows that $a_n > 10^{n-1}$ (for $n$ large enough).