It sounds like you have a cold along with a nosebleed. The common cold
is indeed common this time of year. Cancer presenting in this manner
would be decidedly uncommon. I wouldn't lose any sleep over that at
this point, certainly. Dry winter air causes the lining of your nose
(and the tiny blood vessels within) to become more easily damaged by
blowing your nose, etc. You don't have to blow "hard" to cause a little
bleeding. In some cases, the blood can drain into your throat and thus
your stomach, causing some confusion, especially if someone throws up
and sees blood in their vomitus.

Many people who think they have a "sinus infection" merely have a cold,
which is a viral illness and not treatable with antibiotics. If your
symptoms have persisted for 10 days or more, or they are associated
with a fever, facial *pain* (not merely pressure), or shortness of
breath, then you may have something that warrants a visit to the
doctor. Otherwise, simply try to keep things out of your nose, use a
bit of vaseline petroleum jelly (applied gently with a q-tip) to the
area just inside your nose near the middle part of the nose - the
septum - which will coat the area that is most likely to bleed. Saline
(salt water) nasal spray) can also be used by virtually anyone to help
break up nasal mucus, but use it gently so as not to disloge any clots
and restart the annoying nosebleed.