50 lines
1.4 KiB
TeX
Executable File
50 lines
1.4 KiB
TeX
Executable File
\documentclass[
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solutions,
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hidewarning,
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singlenumbering,
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nopagenumber
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]{../../resources/ormc_handout}
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\usepackage{../../resources/macros}
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\title{Warm-Up: Mario Kart}
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\uptitler{\smallurl{}}
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\subtitle{Prepared by Mark on \today}
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\begin{document}
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\maketitle
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\problem{}
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A standard Mario Kart cup consists of 12 players and four races. \par
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Each race is scored as follows:
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\begin{itemize}
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\item 15 points are awarded for first place;
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\item 12 for second;
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\item and $(13 - \text{place})$ otherwise.
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\end{itemize}
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In any one race, no players may tie.
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A player's score at the end of a cup is the sum of their scores for each of the four races.
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\vspace{2mm}
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An $n$-way tie occurs when the top $n$ players have the same score at the end of a round. \par
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What is the largest possible $n$, and how is it achieved?
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\begin{solution}
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A 12-way tie is impossible, since the total number of point is not divisible by 12.
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\vspace{2mm}
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A 11-way tie is possible, with a top score of 28:
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\begin{itemize}
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\item Four players finish $1^\text{st}$, $3^\text{ed}$, $11^\text{th}$, and $12^\text{th}$;
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\item Four players finish $2^\text{nd}$, $4^\text{th}$, $9^\text{th}$, and $10^\text{th}$;
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\item Two players finish fifth twice and seventh twice,
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\item One player finishes sixth in each race.
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\end{itemize}
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The final player always finishes eighth, with a non-tie score of 20.
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\end{solution}
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\end{document} |