Is being good at mathematic contests necessary to pursue a career in mathematics or physics?

Mathematics takes place at different time-scales. If you can solve a problem in $5$ minutes that others need an hour to solve, you can probably get a good job. If you can solve a problem in a month that others might need a year to solve, you will probably do well as a graduate student. But if you can solve a problem in $10$ years that nobody else can solve in a lifetime, you could be a great mathematician.

It's nice to have awards from contests on your grad school, etc. applications (and it's something to be legitimately proud of), but they have little resemblance to what mathematicians or physicists do in practice. Math is about solving very complicated unsolved problems with months or years of hard work, clever insights, and building off results in the literature; contests are about solving very short contrived problems from scratch with elementary tools under a severe time contraint. It's like asking whether solving crossword puzzles is necessary to be a writer.

Of course, it is not necessary to be good at Olympiads to become a mathematician. Competitions often require to solve a problem in minutes, and don't allow research or collaboration with others. In real mathematics, problems are usually bigger, and the critical factor is not speed, but perseverance and meticulousness.

However, it is not true to say that it will not influence your future. Obviously excelling at an Olympiad looks good for applying to elite schools and securing scholarships. That's not to say you can't get the same benefit otherwise: For instance if you are just not good at being quick, you can instead work hard on your studies, or perhaps even do some kind of research or similar long-term project that shows your abilities.

The second caveat is that while mathematical work in itself is not under as much of a time constraint, becoming a successful mathematician is easier if you do well in school. Doing well in school is often dependent on doing well in exams. Exams are often done with a time pressure. So while speed is not a huge factor for the mathematical work itself, it may be a big factor in getting to that work in the first place.

Lastly, I will dissent from the disdain of those asserting that success in Olympiads mean nothing. They do demonstrate several important things:

• They set a fairly high lower bound on the intelligence of the person. Though clearly you don't need to be a genius to do well at them, having some mental acuity is obviously important.
• They show interest and familiarity with mathematics. Others have pointed out that the problems come down to "knowing the right tricks". Fair enough - with the heavy time constraint you can't do much else anyway. But consider where those tricks would be learned in the first place: One must have some love for mathematics and capacity for independent study to bother learning those tricks at all. Many people would simply be too bored.
• You can only learn so many tricks whilst avoiding mathematical enlightenment. Sooner or later your mind will start to see patterns whether you look for them or not. As such, the quality measured directly in Olympiads is not remarkable. The quality measured by proxy is another matter.

Much like IQ tests, chess and the SAT, by itself the test seems trivial and arbitrary. But it so happens that this arbitrary measure correlates well with success in many areas, and with good reason.

So, to conclude: No, you don't have to be a good Olympian to become a successful mathematician or physicist. But it is not a completely irrelevant measure. You should consider why you are bad at it, whether it's really that hard to just get better, and whether you're willing to accept this unfair handicap (and perhaps be driven to compensate by succeeding in other areas).