10th of Tevet and Jan 1
As Rav Fish mentioned a few weeks ago, this year is the first year that Asara Betevet has fallen out on January 1st since 10 Tevet, 5699/Jan. 1st, 1939.
I'm reminded of the Hasidic idea that from Rosh Hashana until New Years Day, our Rosh Hashana prayers may not have been accepted due to the prosecuting angels prosecuting against us - until January 1st, when the difference is clear: the non-Jews revel with frivolity while the Jewish New Year was done with awe of G-d, all those Heavenly accusations against us disappear and our prayers are accepted.
So this year especially, since we are spending January 1st fasting over the beginning of the Hurban, may our prayers from both Rosh Hashana and Asara Betevet be accepted, and may the Tzom Ha'asiri soon turn into a joyous day for all of Kelal Yisrael.