MathDB

Let

f(x)

be a nonconstant real polynomial. The sequence

\{a_i\}_{i=1}^{\infty}

of real numbers is strictly increasing and unbounded, as

a_{i+1}<a_i+2020.

The integers

\lfloor{|f(a_1)|}\rfloor

\lfloor{|f(a_2)|}\rfloor

\lfloor{|f(a_3)|}\rfloor

\dots

are written consecutively in such a way that their digits form an infinite sequence of digits

\{s_k\}_{k=1}^{\infty}

(here

s_k\in\{0, 1, \dots, 9\}

). If

n\in\mathbb{N}

, prove that among the numbers

\overline{s_{n(k-1)+1}s_{n(k-1)+2}\cdots s_{nk}}

, where

k\in\mathbb{N}

, all

n

-digit numbers appear.

P6