There are many U.S. records that ask for birth date or age. A typical problem that I have found is that the birth dates and ages given in earlier records are often incorrect. My (U.S.-born) father once told me that stating one's exact age or birth date just wasn't important to Lithuanian (or other European) immigrants at that time; he said that it wasn't until the Social Security Act was passed in 1935 that it became important to give (and to know) one's correct birth date (particularly with respect to applying for a Social Security Account Number), and that the mindset of Lithuanians in Chicago changed after that. I don't know if all of that is true, but it's certainly consistent with what I've found in the U.S. records of Lithuanian immigrants.
The U.S. records that asked for a specific birth date are:
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