Display a value if the date value falls within a month

Hello,
Please see the attached example file. I want to display "Y" in cell B2 if 33 days after the date entered in cell A1 falls between 10/1/2015 and 10/30/2015.

If I plug in 9/17/2015 in A1, 33 days after that will be 10/20/2015 which a value between 10/1/2015 and 10/30/2015.

I came up with the following formula:
=IF(AND((A1+33)>="10/1/2015",(A1+33)<="10/30/2015"),"Y","N")

However, instead of displaying "Y" as the result, I keep seeing "N". I am not sure what is wrong?
Very much appreciate your help in advance.

Try it this way

=IF(AND((A1+33)>=DATEVALUE("10/1/2015"),(A1+33)<=DATEVALUE("10/30/2015")),"Y","N")