2023 AMC 12A 第 13 题

Question

在一场乒乓球锦标赛中，每位参赛者都与其他每位参赛者恰好比赛一次。虽然右手选手的人数是左手选手的两倍，但左手选手赢的场数比右手选手赢的场数多

40\%

。（没有平局，也没有双利手选手。）总共进行了多少场比赛？

In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was

40\%

more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?

15

36

45

48

66

答案：B

知识点：组合整除性

难度评级：1660

小提示：

设有

L

名左手选手和

2L

名右手选手，所以总人数为

3L

。

Let there be

L

left-handed and

2L

right-handed players, so

3L

players total

大提示：

若右手选手赢

R

场，则总场数为

2.4R

，所以总场数必须是

12

的倍数

If right-handers win

R

games, the total is

2.4R,

so the total must be a multiple of

12

解答：

设有

L

名左手选手和

2L

名右手选手，则共有

3L

人，总场数为

\binom{3L}{2}

。

若右手选手赢

R

场，则左手选手赢

1.4R

场，所以总场数为

2.4R=\tfrac{12}{5}R

。要使它是整数场数，总场数必须是

12

的倍数。

左手选手至多能赢下所有至少有一名左手选手参加的比赛，即

\binom{L}{2}+2L^2=\tfrac{5L^2-L}{2}

场。另一方面，他们必须赢

\tfrac{7}{12}\binom{3L}{2}=\tfrac{7L(3L-1)}8

场。比较这两个量可得

21L-7\le 20L-4

，所以

L\le 3

。

依次试

L=1,2,3

，总场数为

3,15,36

；其中只有

36

是

12

的倍数，且可实现：

3

名左手选手可以赢下全部

18

场混合对局以及左手内部的

3

场，共

21=1.4\cdot 15

场。

所以正确答案是 B。

Let there be

L

left-handed and

2L

right-handed players, for

3L

players and

\binom{3L}{2}

games total.

If right-handers win

R

games, left-handers win

1.4R,

so the total is

2.4R=\tfrac{12}{5}R.

For this to be an integer count, the total number of games must be a multiple of

12.

The left-handers can win at most every game involving at least one left-hander, namely

\binom{L}{2}+2L^2=\tfrac{5L^2-L}{2}.

On the other hand, they must win

\tfrac{7}{12}\binom{3L}{2}=\tfrac{7L(3L-1)}8

games. Comparing these quantities gives

21L-7\le 20L-4,

so

L\le 3.

Testing

L=1,2,3

gives totals

3,15,36;

only

36

is a multiple of

12.

It is achievable: the

3

left-handers can win all

18

mixed games and their

3

internal games, giving

21=1.4\cdot 15

wins.

Thus, the correct answer is B.

2023 AMC 12A 第 13 题

答案：B

难度评级：1660

小提示：

大提示：

解答：

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