2006 SMT/Advanced Topics Problems/Problem 6

Problem

Ten teams of five runners each compete in a cross-country race. A runner finishing in $n\text{th}$ place contributes $n$ points to his team, and there are no ties. The team with the lowest score wins. Assuming the first place team does not have the same score as any other team, how many winning scores are possible?

Solution

Notice that the smallest possible winning score is if the five runners get first, second, third, fourth, and fifth, and the score would be $1+2+3+4+5$ . Now let $n$ be the largest possible winning score. Since there are no ties, the minimum score for the other teams is $n+1$ , so the minimum value for the sum of all of the scores is $n+9(n+1)=10n+9$ . However, we know that the sum of all of the scores must be the sum of all integers from $1$ to $50$ , which is $\frac{50\cdot51}{2}=1275$ . Therefore, $10n+9\le1275\implies n\le126.6$ , so the largest possible winning score is $126$ . Notice that all scores in between are also possible winning scores, and they can be achieved by simply switching two places. Therefore, the number of possible winning scores is $126-15+1=\boxed{112}$ .

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2006 SMT/Advanced Topics Problems/Problem 6

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