Question

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?

$15$

$36$

$45$

$48$

$66$

Solution:

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 $12.$

Testing $L=1,2,3$ gives totals $3,15,36;$ only $36$ is a multiple of $12,$ and it is achievable (the $3$ left-handers can take all $18$ mixed games plus their $3$ internal games for $21=1.4\cdot 15$ wins).

Thus, the correct answer is B.

2023 AMC 12A Problem 13

Solution:

Problem 13 in Other Years