Are participations in competitions well recognized for a Ph.D. application?
Competitions can be valuable evidence of achievement, but they have to be not just widely recognized, but also really relevant to the field.  (Nobody cares if you're a chess champion, since you aren't going to grad school in chess, and the fact that it involves hard work and talent will not help your case; if anything, it will be viewed as a potential distraction that may cause trouble in the future.)  The best case is if some of the faculty once participated in the same competition.  Then they will know exactly what's involved and what success means.  The next best case is if some of their current students participated.  Otherwise, it will mean very little, unless your recommenders somehow make a strong case for its importance.

As a test case, let's think about the Putnam examination, which is the most prestigious math contest for undergraduates in the US and Canada.  Doing well on the Putnam exam is very valuable in math grad school applications, but even being one of the winners is not a guarantee of admission.  The big advantage of the Putnam is that it gives objective evidence of talent compared with a nationwide pool, but the disadvantage is that solving contest problem is really not the same thing as doing research.  If an undergraduate writes a paper a faculty member would really be proud of having written, then it looks better than winning the Putnam exam, but most undergraduate papers do not rise to that level and may not be as impressive as winning the exam.  It's hard to quantify this trade-off, but I would definitely not advise anyone to neglect research opportunities in order to prepare better for the Putnam.  Ultimately, graduate school is about research, and admissions committees sometimes worry about applicants who look more interested in competitions than research.

The Mathematical Contest in Modeling may be more along the lines of the contests you mention: it's a multi-day, team-based contest.  My impression is that it carries less weight in admissions decisions than the Putnam exam does.  Being able to help organize and train a winning team has real value, but it won't play much of a role in graduate admissions.