I am using version Rel. 5.2.0, September 15th, 2018 on OSX but get the same answers on other platforms with other versions.
prompt> echo 3 0 | proj +proj=utm +zone=1 +ellips=wgs84 500000.00 19995929.89
Seems correct for the easting as the center of the zone should be 500km. The northing decreases to about 10km for a value of 90 degrees latitude as shown below.
prompt> echo 3 90 | proj +proj=utm +zone=1 +ellips=wgs84 500000.00 9997964.94
However, when I give a sightly negative latitude I get
prompt> echo 3 -.0001 | proj +proj=utm +zone=1 +ellips=wgs84 500000.00 -19995918.83
Which means that (3,0) and (3,-.0001) have very, very different coordinates in the northern hemisphere utm zone 1. I don't think this is correct -- hopefully, I am simply wrong and have made an error somewhere.