Problem:
What is the value of
(1−321​)(1−521​)(1−721​)​(1+31​)(1+51​)(1+71​)​?
Answer Choices:
A. 3​
B. 2
C. 15​
D. 4
E. 105​
Solution:
Let x be the displayed expression. Then
x2​=(1−321​)(1−521​)(1−721​)(1+31​)2(1+51​)2(1+71​)2​=(1+31​)(1+31​)​⋅(1−31​)(1+31​)​⋅(1+51​)(1+51​)​⋅(1−51​)(1+51​)​⋅(1+71​)(1+71​)​⋅(1−71​)(1+71​)​=(1−31​)(1+31​)​⋅(1−51​)(1+51​)​⋅(1−71​)(1+71​)​=24​⋅46​⋅68​=4.​
Therefore x=(B)2​.
The problems on this page are the property of the MAA's American Mathematics Competitions