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

A B

,

B C

,

C D

,

D E

, and

E A

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

Accepted Answer

Let a be the number placed at A, let b be the number placed at B and so on, so a, b, c, d, e are equal to 3, 5, 6, 7, 9, in some order. Let v, w, x, y, z be the arithmetic sequence. In the sum v + w + x + y + z, each of a, b, c, d, e is counted twice, so v + w + x + y + z = 2(a + b + c + d + e) = 2(3 + 5 + 6 + 7 + 9) = 2 30 = 60.Therefore, the middle term x is 605 = 12. The diagram below shows a possible arrangement. FIGURE