Problem:
Let
x=(5​+1)(45​+1)(85​+1)(165​+1)4​
Find (x+1)48.
Solution:
Let y=165​. Then
x=(y8+1)(y4+1)(y2+1)(y+1)4​​=(y8+1)(y4+1)(y2+1)(y+1)(y−1)4(y−1)​=y16−14(y−1)​=5−14(y−1)​=y−1​
Thus (x+1)48=y48=53=125​.
The problems on this page are the property of the MAA's American Mathematics Competitions