Translation:Nice to meet you, my name is Maria.
申す is the humble (謙譲語 - けんじょうご - kenjougo) verb for "speak, say" , making this an especially polite way to speak to another, typically reserved for social superiors (such as an employee addressing a senior company manager).
At its most literal it's something like "have the honour to do / be", and is also used in with nouns in the sense of "I have the honour / I humbly [do noun]".
I put "How do you do." for this but it was counted as incorrect. Hajimemashite can be translated as "It is nice to meet you" and "How do you do". Both are correct. Your app says that "hello" is the answer. However, hajimemashite is used when meeting for the first time only. It doesn't mean "hello". Therefore, my answer is actually correct and must be acceptable!
In this context, Hajimemashite are used as a greeting. Such as hello, good morning, etc. When greeting a person, generally when u say hello the other person will reply hello as well. The same with hajimemashite, generally the other person will answer hajimemashite as well. "Nice to meet you" "ah nice too meet you too".
"How do you do" are a different in that aspect, as it question the other person condition. While it's acceptable as a form of greeting, it uses are kinda not the same as hajimemashite. Normally it would be weird if someone ask "how do you do" and the other person reply with "how do you too".
Check this out. Best explanation yet! http://yesjapan.com/YJ6/question/4125/when-introducing-yourself_which-is-better-to-use-desu_to-moushimasu_or-to-iimasu
Yes, I was so very thrown off as well. I'd actually never even heard of と申します (ともうします). Here is a simple explanation: http://yesjapan.com/YJ6/question/4125/when-introducing-yourself_which-is-better-to-use-desu_to-moushimasu_or-to-iimasu
This is more formal. Also, in Japanese if you've already highlighted yourself as the subject (e.g someone asks what your name is) then it's seen as rude to reinstate that you are the subject. Think of it like cutting in when someone's speaking to talk about yourself. I hope this helped :)
The expected answer is "nice to meet you", "hello" is just one of the variants that is accepted. If you think "hello" shouldn't be accepted you can report the sentence when you come across it, but reporting things in the sentence discussions themselves is unlikely to be noticed by anyone who can do anything about it.
1) 初めまして is not normal kanji usage. It is usually written as はじめまして. It's not wrong to use the kanji, but it's unusual, so maybe you just need to submit an error report so it will be added to the database.
2) It was a "type what you hear" question. Duolingo's programming was designed for languages that only have one way of spelling something, while Japanese has multiple ways. Unfortunately, those questions only accept one specific combination of kanji and kana as a "correct" answer, even though there are multiple correct answers.