2005 AMC 12A Problem 13

Question

In the five-sided star shown, the letters

A,

B,

C,

D,

and

E

are replaced by the numbers

3,

5,

6,

7,

and

9,

although not necessarily in that order. The sums of the numbers at the ends of the line segments

AB,

BC,

CD,

DE,

and

EA

form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?

9

10

11

12

13

Each of the five numbers is an endpoint of exactly two segments

The middle term of a five-term arithmetic sequence equals the mean of all five terms

Every number appears as an endpoint of exactly two of the five segments, so the total of the five sums is

2(3 + 5 + 6 + 7 + 9) = 60.

The middle term of a five-term arithmetic sequence is its mean, namely

60 \div 5 = 12.

Thus, the correct answer is D.

Problem 13 in other years