Can a result of (any) hash algorithm contain the hash result itself? [closed]

Question

Suppose you have a file of 240 lines. Any lines, any content. You then calculate the hash of that file, let's say MD5, and the result is something in the following structure:

f8dbe310c1f61066d766071b07503ce8

Now, you take this hexadecimal digest, and add it to the file as line 241.

Rehashing the file (with additional hash line) will result with a new hash value obiously.

My questoin is whether it is even theoretically possible for a hash value to contain itself, meaning, can there be a file, with some content that its hashed value is identical to a string already presented in the file (e.g. line 241 in this example)

Answer 1

For the first question, about the last line, it surely depends on the hash function. For example, suppose each line is a single bit (0 or 1). If the hash function is the xor of the bits, then the answer is yes -- take 240 lines with total parity 0. But the answer is no if the hash function is defined by $$f(x_1, x_2, \ldots, c_n) = \neg (x_n \oplus f(x_1, \ldots, x_{n-1})),$$ with $f()=0$, where $\oplus$ is xor.

For the second question (about any line), I think @Yonatan's point in his comment was that if the hash function has finitely many possible outputs (as e.g. MD5 does), then, for a file that already contains every possible hash output (one per line), no matter what the hash outputs it will be in the file already.