Question

Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers $1$ through $10.$ Each throw hits the target in a region with a different value. The scores are: Alice $16$ points, Ben $4$ points, Cindy $7$ points, Dave $11$ points, and Ellen $17$ points. Who hits the region worth $6$ points?

$\text{Alice}$

$\text{Ben}$

$\text{Cindy}$

$\text{Dave}$

$\text{Ellen}$

Solution:

The only way to get Ben's score is with a $1$ and $3$ since he can't hit $2$ twice.

Cindy can achieve her score with $1 + 6, 2 + 5, \text{ or } 3 + 4.$ Ben already hit $1$ and $3,$ so Cindy must have hit $2$ and $5.$

Similarly, Dave must have hit $4$ and $7.$ Finally, since $7$ is already used, Alice is forced to have hit $6$ and $10$ with Ellen hitting $8$ and $9.$

Thus, A is the correct answer.

2004 AMC 8 Problem 18

Solution:

Problem 18 in Other Years