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Problem 734

AMC 10/12, early questions

Combinatorics Difficulty 3.5 Multiple choice CEMC Pascal · Canada · 2022

Alvin, Bingyi and Cheska play a two-player game that never ends
in a tie. In a recent tournament between the three players, a total of
60 games were played and each pair of players played the same number of
games.

When Alvin and Bingyi played, Alvin won 20% of the
games.
When Bingyi and Cheska played, Bingyi won 60% of the
games.
When Cheska and Alvin played, Cheska won 40% of the
games.

How many games did Bingyi win?

Pick one

Official solution

Since 60 games are played and each of the 3 pairs plays the same
number of games, each pair plays $60 \div 3 = 20$ games.

Alvin wins 20% of the 20 games that Alvin and Bingyi play, so Alvin wins
$\frac{20}{100} \times 20 = \frac{1}{5} \times 20 = 4$ of these 20 games and Bingyi wins $20 - 4 = 16$ of these 20 games.

Bingyi wins 60% of the 20 games that Bingyi and Cheska play, so Bingyi
wins a total of $\frac{60}{100} \times 20 = \frac{3}{5} \times 20 = 12$ of these 20 games.

The games played by Cheska and Alvin do not affect Bingyi’s total
number of wins.

In total, Bingyi wins $16 + 12 = 28$
games.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.