jmc

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 this order. The sums of the numbers at the ends of the line segments

\overline{A B}

,

\overline{B C}

,

\overline{C D}

,

\overline{D E}

, and

\overline{E A}

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

Accepted Answer

Each number appears in two sums, so the sum of the sequence is

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

The middle term of a five-term arithmetic sequence is the mean of its terms, so

60/5 = 12

is the middle term.

The figure shows an arrangement of the five numbers that meets the requirement.