2013/14 AHSMC Part 1 Problem 4

Problem 4 of 16

Two teams $A$ and $B$ played a soccer game on each of seven days. On each day, the first team to score seven goals won. There were no ties. Over the seven days, $A$ won more games than $B$ , but $B$ scored more goals than $A$ overall. The difference in the total numbers of goals scored by $B$ and $A$ is at most

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Solution

$B$ won at most three games, and for each of these games, $B$ wins by at most $7$ goals with a $7$ to $0$ score. In the other four games, $B$ loses by at least $1$ goal with a $7$ to $6$ score. The goal difference is at most $7 \times 3 - 1 \times 4 = 17$ . The answer is (A).

University of Alberta

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