\documentclass[
	solutions,
	hidewarning,
	singlenumbering,
	nopagenumber
]{../../resources/ormc_handout}
\usepackage{../../resources/macros}


\title{Warm-Up: Mario Kart}
\uptitler{\smallurl{}}
\subtitle{Prepared by Mark on \today}


\begin{document}

	\maketitle

	\problem{}
	A standard Mario Kart cup consists of 12 players and four races. \par
	Each race is scored as follows:
	\begin{itemize}
		\item 15 points are awarded for first place;
		\item 12 for second;
		\item and $(13 - \text{place})$ otherwise.
	\end{itemize}
	In any one race, no players may tie.
	A player's score at the end of a cup is the sum of their scores for each of the four races.

	\vspace{2mm}

	An $n$-way tie occurs when the top $n$ players have the same score at the end of a round. \par
	What is the largest possible $n$, and how is it achieved?

	\begin{solution}
		A 12-way tie is impossible, since the total number of point is not divisible by 12.

		\vspace{2mm}

		A 11-way tie is possible, with a top score of 28:
		\begin{itemize}
			\item Four players finish $1^\text{st}$, $3^\text{ed}$, $11^\text{th}$, and $12^\text{th}$;
			\item Four players finish $2^\text{nd}$, $4^\text{th}$, $9^\text{th}$, and $10^\text{th}$;
			\item Two players finish fifth twice and seventh twice,
			\item One player finishes sixth in each race.
		\end{itemize}
		The final player always finishes eighth, with a non-tie score of 20.
	\end{solution}

\end{document}