During undergrad, students can face several opportunities, ranging from participation in (1) research groups, (2) study/training for competitions, (3) internships, (4) opening start-ups with colleagues (the latter is becoming very common these days), but I wonder, what is really important, for a Ph.D. application/admission...
By competition I mean ones like: INTEL GLOBAL CHALLENGE (VC), ACM ICPC, IMAGINE CUP (examples from the Comp. Sci. field, and Business, but indeed there may be a bunch of these in other fields, that I don't know)
Talking about (1) and (2) aforementioned...
If a student stays a long time during his/her undergrad in a (world-class) research group, he/she is likely to have the opportunity to publish a bunch of papers (some of these might be good, well-referenced, etc; some of these might be not as good), and meet some good researchers around the world, and so on; it really required dedication.
On the other hand, take part into a training class for competitions (that requires dedication, as well) may lead students to gains in terms of working in group, time-boxed activities, etc, as well as to face the opportunity of proposing solutions for real-world problems, and so on.
It is really tough to do both, in order to have great results, since in both cases time and dedication is mandatory. Indeed, there are some "outliers" students that can do both in a very good way, but I'm trying to generalize my assumption, by considering "average" students.
Hence, it's known that good papers have a great influence on the Ph.D. admission (despite of other well-known recommendations, e.g., a great GRE, good recom. letters, and so on), but I'd like to know if (and how) the universities consider students who dedicated their time to join this kind of competitions, obtaining some prizes, as a consequence, etc.
Answer
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.
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