Problem:
Suppose that 4x1=5,5x2=6,6x3=7,…,127x124=128. What is x1x2⋯x124 ?
Answer Choices:
A. 2
B. 25
C. 3
D. 27
E. 4
Solution:
Since 4x1=5,5x2=6,…,127x124=128, we have
47/2=128=127x124=(126x123)x124=126x123⋅x124=⋯=4x1x2⋯x124.
So x1x2⋯x124=7/2.
\section*{OR}
We have
x1x2⋯x124=log45⋅log56⋯log127128=log4log5⋅log5log6⋯log127log128=log4log128=log22log27=2log27log2=27.
The problems on this page are the property of the MAA's American Mathematics Competitions