diff --git a/Misc/Warm-Ups/mario.tex b/Misc/Warm-Ups/mario.tex new file mode 100755 index 0000000..7e664b8 --- /dev/null +++ b/Misc/Warm-Ups/mario.tex @@ -0,0 +1,49 @@ +\documentclass[ + solutions, + hidewarning, + singlenumbering, + nopagenumber +]{../../resources/ormc_handout} +\usepackage{../../resources/macros} + + +\title{Warm-Up: Mario Kart} +\subtitle{Prepared by \githref{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} \ No newline at end of file