All three are correct!

Somewhere in this world, someone might be facing the same problem as (1) me (2) mine (3) I am. Let’s try the first : (1) Somewhere in this world, someone might be facing the same problem as me. That means someone might be facing the same problem as me facing it : this is a ‘’heavy’’ sentence, it sounds incorrect. (2) Somewhere in this world, someone might be facing the same problem as mine. That means someone might be facing the same problem as ‘’ my problem’’; so the point is made on the problem itself that someone could face. It’s correct. (3) Somewhere in this world, someone might be facing the same problem as I am. That means someone might be facing the same problem as I am ‘’facing this problem’’ ; the focus is on the ACTION that I am doing, I compare the action that I am doing with the same action someone will do . it’s correct . Here is some other rules, to decide whether to use the subject or object pronoun after the words than or as, mentally complete the sentence. Examples: 1- Mia_lia is as smart as she/her. If we mentally complete the sentence, we would say, " Mia_lia is as smart as she is." Therefore, she is the correct answer. 2- Jura is taller than I/me. Mentally completing the sentence, we have, "Jura is taller than I am." 3- Eliot would rather talk to her than I/me. We can mentally complete this sentence in two ways: "Eliot would rather talk to her than to me." OR "Eliot would rather talk to her than I would." As you can see, the meaning will change depending on the pronoun you choose. Hope I helped

I accept all three as correct, but I recently discovered a old-ish rule where comparatives do not take an object form, which makes "...as me" incorrect. I suppose the theory is that between the two things being compared, they are still on an equal footing grammatically (one does not actually affect the other), giving you two subjects in the sentence. So, "the same problem as I am (facing...)" is the correct answer. Apparently. I'll add that this rule is often hotly debated or outright ignored. :)

I agree with Eliot

All correct