Question

Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party?

$1$

$4$

$5$

$6$

$7$

Solution:

Sarah invites her friends and the classmates who are friends with at least one of her friends. In graph terms, she invites dots that are $1$ or $2$ line segments away from Sarah.

From the graph, $4$ dots are disconnected from Sarah’s component, and $2$ more dots in Sarah’s component are $3$ segments away.

Those $4+2=6$ classmates will not be invited.

Thus, D is the correct answer.

2003 AMC 8 Problem 18

Solution:

Problem 18 in Other Years