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As a citizen of the US you would need to report your worldwide income. This includes interest earned on a savings or checking account anywhere.
If one of your accounts earns interest and that is the only income at all that you receive then you would look to your filing requirement. If you are a single person and your have $11,700 of income you have to file a US tax return. If you do not have that much then you are not required to file and would not pay tax on the income. States have different amounts for requirements.
If your father is not a US person and he sends you money or makes gifts to you, you have to report that if the amount or value is $100,000 or more in the year. The form is 3520 to use to report and this is even if you are not filing a tax return. The gift itself is not taxable but reporting is required. This includes the cash and a house he would by. No tax on the gift but reporting is a must.
All US citizens are required to report their worldwide income. If your mother doe snot have any income then she would not be required to file a tax return but if she has ant accounts in her name (or jointly) then she may have to file a reporting form listing those accounts.
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