File containing its own checksum
I created a piece of code in C, then ran bruteforce for less than 2 minutes and got this wonder:
The CRC32 of this string is 4A1C449B
Note the must be no characters (end of line, etc) after the sentence.
You can check it here: http://www.crc-online.com.ar/index.php?d=The+CRC32+of+this+string+is+4A1C449B&en=Calcular+CRC32
This one is also fun:
I killed 56e9dee4 cows and all I got was...
Source code (sorry it's a little messy) here: http://www.latinsud.com/pub/crc32/
Yes. It's possible, and it's common with simple checksums. Getting a file to include it's own md5sum would be quite challenging.
In the most basic case, create a checksum value which will cause the summed modulus to equal zero. The checksum function then becomes something like
(n1 + n2 ... + CRC) % 256 == 0
If the checksum then becomes a part of the file, and is checked itself. A very common example of this is the Luhn algorithm used in credit card numbers. The last digit is a check digit, and is itself part of the 16 digit number.