sure.... there is probably an easier way, but this will work for now:
<?php
$day = 10; // Change this as needed to whatever day. Just using yoru example.
$today = strtotime("now");
$mon = (date('d')<$day)?date('m'):date('m')+1;
$mon = ($mon==13)?1:$mon;
$year = ($mon!=date('d') && date('m')==12)?date('Y')+1:date('Y');
$next = mktime(12,0,0, $mon, $day, $year);
$dif = $next-$today;
$days = floor($dif/(60*60*24));
echo 'There are '.$days.' days remaining until the '.$day;
?>
THere are probably much simpler ways, but that works.
~Brett