Problem:
All the numbers are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Suppose that one pair of opposite faces of the cube are assigned the numbers and , a second pair of opposite faces are assigned the numbers and , and the remaining pair of opposite faces are assigned the numbers and . Then the needed sum of products is ace . The sum of these three factors is . A product of positive numbers whose sum is fixed is maximized when the factors are all equal. Thus the greatest possible value occurs when , as in . This results in the value .
The problems on this page are the property of the MAA's American Mathematics Competitions