I would like to know if you can say "我好", when someone asks you "你好吗?".
I am aware you can answer "我很好" as in "I am very good", but can you say "I am good"? Please fix my mistakes if I have any.
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We never answer “我好” when asked “你好吗?”. Indeed, these days it's not very common for people to greet others with “你好吗？” face to face, but simply “你好” will do just fine. When greeted with “你好”, we can acknowledge it with the same “你好”. 我很好 sometimes means I am ok (so don't worry about me). Imagine two lovers just broke up but still treat each other as friends. In their letters later on, one may write 我很好，不用担心我. This is the scenario that popped into my head when I first saw this quesiton.
You must be very cautious about translating 很 in a hyper-literal way. In Mandarin it often functions ONLY to link a noun and adjective, and consequently does not necessarily add the literal sense of "very". Hence your possible reluctance to use "我很好", even though it is the appropriate response for what you want to express, is unwarranted.
This point is mentioned here.
Sometimes you need to think 很 as a linking verb, in English you never say:
but: I am fine.
The soup delicious.
but: The soup tastes delicious.
The same in this case, 很 as a linking verb, you can't omit it.
我 很 好 = I am fine. (not "very well"), there is no "very" involved.
But if you want to express "I am very/quite well", you need to say:
我 非常 好. (that important note is now you don't need 很 anymore, since 非常 already carried the meaning of BE.)
I think it is not correct to translate "我很好" into "I am very good". It means "I am doing fine", "I am doing well". When one say "你好吗？"，“好” is an adverb, means "How are you doing"，"Are you doing OK"？
In a less satisfiction case, say “我还好”，"还行吧"，“就那样吧”，“还是那样”。
Why do not say "我好"？Because it is sound like the adjective "good". People do not use it to themselves.