Why are you using the parameter parent=1?
If you remove this parameter it should work.
If you want children and grandchildren, you should use:
$categories = get_categories('hide_empty=0&child_of=".$cat);
If you want only direct children, you should use:
$categories= get_categories("hide_empty=0&parent=".$cat);
Also note that you are echoing $mycat inside the loop, and you are joining the multiple categories in this var, so the categories should appear repeated…
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