1989 AMC 12 Problem 25

Question

In a certain cross-country meet between two teams of five runners each, a runner who finishes in the

n

th position contributes

n

to his team’s score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?

10

13

27

120

126

The two team scores add to

1+2+\cdots+10

Determine the minimum winning score and verify that every integer below half the total is attainable

The two scores sum to

55,

so the winning score is at most

27.

Its minimum is

1+2+3+4+5=15.

Every score from

15

through

20

is obtained by

\{1,2,3,4,k\}

for

k=5,\ldots,10.

Scores

21

through

25

use

\{1,2,3,k,10\}

for

k=5,\ldots,9,

and

26,27

use

\{1,2,4,9,10\}

and

\{1,2,5,9,10\}.

Thus all

13

integers from

15

through

27

are possible.

Thus the correct answer is B.

Problem 25 in other years