42 lines
1.2 KiB
Typst
42 lines
1.2 KiB
Typst
#import "@local/handout:0.1.0": *
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#show: doc => handout(
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doc,
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quarter: link(
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"https://betalupi.com/handouts",
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"betalupi.com/handouts",
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),
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title: [Warm-Up: Mario Kart],
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by: "Mark",
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)
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#problem()
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A standard Mario Kart cup consists of 12 players and four races. \
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Each race is scored as follows:
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- 15 points are awarded for first place;
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- 12 for second;
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- and $(13 - #text("place"))$ otherwise.
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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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#v(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. \
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What is the largest possible $n$, and how is it achieved?
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#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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#v(2mm)
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A 11-way tie is possible, with a top score of 28:
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- Four players finish $1^#text("st")$, $3^#text("ed")$, $11^#text("th")$, and $12^#text("th")$;
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- Four players finish $2^#text("nd")$, $4^#text("th")$, $9^#text("th")$, and $10^#text("th")$; // spell:disable-line
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- Two players finish fifth twice and seventh twice,
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- One player finishes sixth in each race.
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The final player always finishes eighth, with a non-tie score of 20.
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])
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