Problem:
The sums of three whole numbers taken in pairs are 12,17, and 19. What is the middle number?
Answer Choices:
A. 4
B. 5
C. 6
D. 7
E. 8
Solution:
Let the three whole numbers be a<b<c. The set of sums of pairs of these numbers is (a+b,a+c,b+c)=(12,17,19). Thus 2(a+b+c)= (a+b)+(a+c)+(b+c)=12+17+19=48, and a+b+c=24. If follows that (a,b,c)=(24−19,24−17,24−12)=(5,7,12). Therefore the middle number is (D)7​.
The problems on this page are the property of the MAA's American Mathematics Competitions