Problem:
Let x=44n=1. What is the greatest integer that does not exceed 100x?
Solution:
Because sinn∘+cosn∘=2cos(45−n)∘, it follows that
n=1∑44sinn∘+n=1∑44cosn∘=n=1∑442cos(45−n)∘=n=1∑442cosn∘
Thus
n=1∑44sinn∘=(2−1)n=1∑44cosn∘
which yields x=1+2 and ⌊100x⌋=241.
The problems on this page are the property of the MAA's American Mathematics Competitions