Do You Really Need to Win Math Contests to be a Mathematician?

No.

I just wanted to follow up on the previous posts about the recent IMO contest.

Reasoning by anecdotes is tricky in general, but single counterexamples suffice for disproving statements with "All" quantifiers. 

I've known many successful mathematicians with no history of success doing contest math, and I knew an absolute Olympiad phenom who was not a successful mathematician.

Therefore skill in contest math is neither necessary nor sufficient.

That said, in my experience, success in hard math contests, is usually an indication of some combination of talent, an excellent education, and specialized training which seems to make later success in math more likely. In addition to all of the extra training, succeeding on these contests early will give you access to even more training, resources, and opportunities, so there is a little bit of "the rich get richer" compounding effect going on. It would be interesting to try and quantify these assertions but I will leave that to someone else.

As an aside, the Olympiad phenom mathematician who I knew as an undergraduate at UIC, was Norman Hamilton.  My first mentor and neighbor, Victor Gugenheim. introduced me to him.  Paul Halmos in his book "I Want to Be a Mathematician: An Automathography", says the following about Norman:

"Princeton University still had an important role in the life of the Institute mathematicians. I had two good friends among the students: Norman Hamilton (about 18 or 19 then still an undergraduate, flunking courses in Latin for non-attendance, and doing brilliantly in graduate courses in mathematics), and Oscar Goldman (later chairman at the University of Pennsylvania; at the time a comparatively elderly 21 or 22)."

Norman then went to graduate school at U of C and things didn't go that well which is VERY disturbing in my opinion, but goes to show that it isn't just underrepresented students whose talent and potential is wasted...just mainly.

You could give Norman any Putnam level problem and he would solve it...usually quickly. It was easy to assume that he had just seen them all, but you could give him obscure contest problems too and the same thing would happen. Unfortunately, he would solve them in between puffs on his Camel cigarettes so you had to be willing suffer through the cancer risk of second hand smoke -- which I did.

That's right I was willing to risk death to learn math! I like to contrast this with many modern students who are frequently not even willing to risk failure in my experience. There was another student who was also hanging on Norman's every word, but he was already a smoker.

I can't say this amounted to real Putnam training, but this experience inspired me to sign up for the Putnam. I almost immediately got intimidated and my brain locked up, and I then spent the next 2.5 hours doodling. The only reason I stayed for the entire first half was that a free lunch at Manny's was being offered and I like deli food. I went home instead of staying for the second half. That was the only time I ever "took" the Putnam and concluded that ARML was as far as I was going to go in the world of contest math.

I used to always tell my former students, who were interested in math contests this story, since I think it is important to let students know that you are not a genius when you are not actually a genius, and also that they shouldn't get too hung up on math contests. If you like them, they are fun, and that's about it.

Addendum 1: Did you notice the comment from Halmos, "a comparatively elderly 21 or 22" ? It's funny, but mathematicians love to tell you you're too old in my experience.

Addendum 2: This Oscar Goldman guy is quite impressive! After his math career I think he may have moved to intelligence work running an agent named Steve Austin, a man barely alive.