Although competition math is very different from research math, performance in math competitions is still very much positively correlated with research potentials. Many top researchers were top competitors in math competitions when they were young. Just for an example, among the most recent fields medalists, 3 out of 4 had excellent performances in IMO (Stanislav Smirnov and Ngô Bảo Châu both had perfect scores). And for 2006 fields medals, there were famous examples like Terence Tao and Perelman.
I believe some professors will take things like Putnam, IMO, IMC into consideration seriously (and surely some may not). Having said that, definitely recommendation letters, research experience and grades are all much more important. But if you are a marginal candidate (say, the admission committee can only admit one of you and another applicant who had similar profile as yours, then the competition results may make you preferred).
In any case, it won't hurt you at least.