\documentclass[11pt]{scrartcl}
\usepackage{evan}
\begin{document}
\title{Shortlist 1998 N8}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 59}
\maketitle
\section*{Problem}
Let $a_{0}$, $a_{1}$, $a_{2}$, \dots\ be an increasing sequence of nonnegative integers
such that every nonnegative integer can be expressed
uniquely in the form $a_{i}+2a_{j}+4a_{k}$,
where $i$, $j$ and $k$ are not necessarily distinct.
Determine $a_{1998}$.
\section*{Video}
\href{https://www.youtube.com/watch?v=46z5jJ-rauc&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/46z5jJ-rauc}}
\section*{External Link}
\url{https://aops.com/community/p124444}
\newpage
\section*{Solution}
It is clear by induction there is at most one sequence,
since at any point $a_i$ must be equal to the smallest integer
not expressible using $a_0$ through $a_{i-1}$.
On the other hand, one can give an example of a sequence:
take $\{a_i\}$ as a set to be the numbers
that have only the digits $0$ and $1$ in their octal (base-8)
representation.
Since \[ 1998 = 2048 - 50 = 1984 + 14 = 11111001110_2, \]
it follows the answer is $11111001110_8 = 1{,}227{,}096{,}648$.
\end{document}