AMC8 2023

AMC8 2023 · Q8

AMC8 2023 · Q8. It mainly tests Basic counting (rules of product/sum), Logic puzzles.

Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers $1$ and $0$ represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo’s win-loss record?

Lola、Lolo、Tiya 和 Tiyo 参加了一场乒乓球锦标赛。每位选手与每位其他三位选手各比赛两次。下面显示了选手们的胜负记录。数字 $1$ 和 $0$ 分别代表胜利或失败。例如，Lola 赢了五场比赛，输了第四场比赛。Tiyo 的胜负记录是什么？

(A) \ \texttt{000101} \ \texttt{000101}

(B) \ \texttt{001001} \ \texttt{001001}

(D) \ \texttt{010101} \ \texttt{010101}

(E) \ \texttt{011000} \ \texttt{011000}

Answer

Correct choice: (A)

正确答案：（A）

Solution

We can calculate the total number of wins ($1$'s) by seeing how many matches were players, which is $12$ matches played. Then, we can calculate the # of wins already on the table, which is $5 + 3 + 2 = 10$, so there are $12 - 10 = 2$ wins left in the mystery player. Now, we will make the key observation that there is only $2$ wins ($1$'s) per column as there are $2$ winners and $2$ losers in each round. Strategically looking through the columns counting the $1$'s and putting our own $2$ $1$'s when the column isn't already full yields $\boxed{\textbf{(A)}\ \texttt{000101}}$.

我们可以计算总胜利次数（$1$ 的数量），因为总共有 $12$ 场比赛。然后，计算表格中已有的胜利次数，为 $5 + 3 + 2 = 10$，所以神秘选手剩下 $12 - 10 = 2$ 次胜利。现在，我们做出关键观察：每列有正好 $2$ 个胜利（$1$），因为每轮有 $2$ 个胜者和 $2$ 个败者。通过策略性地查看列中 $1$ 的数量，并在列未满时放置我们自己的 $2$ 个 $1$，得到 $\boxed{\textbf{(A)}\ \texttt{000101}}$。

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