There are many design contests in which to participate, but deciding whether to do so involves many hard-to-assess and personal factors.
Design contests are very popular, and with good reason: The "winner" gets a prize, publicity, and perhaps opportunity to execute the idea. These contests vary in scope from modest ("design a 5-V/1-A power supply with 95%+ efficiency") to very ambitious, such as the SpaceX Hyperloop Design Competition, in which the winning team is being funded to build a prototype, or NASA's Small Electric Airplane Challenge.
I have mixed views about these contests. Some, such as the Hyperloop, will let the winner advance to the next stage with resources that are otherwise unavailable. They also encourage some serious and innovative thinking, backed by solid analysis. Other contests seem to be primarily publicity stunts, which give the contest sponsor access to efforts of others at very low expense, while they spend little. [Note that all contests are different than various innovation awards, such as the highly regarded Breakthrough Awards, which are given without the need for any actual entry or participation.]
Others are excellent exercises for the participants, who would likely be doing something similar as a project anyway, such as the student participating in the annual Texas Instruments Innovation Challenge for students, which features a dazzling array of projects with working circuits, PCB layouts, software, bill of materials, and analysis of what went as planned and what didn't.
Obviously, the individuals and teams who spend their time and energy—and often money—on these design contests have different motives, incentives, and rationales. Some have ideas and time, some look for the publicity (how much you get varies widely), some need to do something anyway, some want the prize money (often less than the costs incurred), and others have the innovation urge and the contest is the final spark that gets them to act on it.
Still, it's an interesting dilemma of who should enter which contests and under what conditions. In many of these situations, entrants sign away all rights to their ideas and innovation, thus giving up intellectual property potential, while the contest sponsor gets the IP at modest or no cost. But it is not a win/lose: Without the contest resources and incentive, the idea(s) might never be explored and even commercialized.
As in most engineering situations and life decisions, there is no simple, single "right" answer. Whether to enter a contest or not is a function of many hard-to-assess variables including time needed, effort required, opportunity cost, out-of-pocket cost, credibility of the judges, judgment metrics, risks, odds of winning, return on investment for non-winners, tangible return on investment for winners, potential for profit (if you care), award of additional resources to go further, and more.
The credibility and integrity of the sponsor of the contest is also a major issue. While many are run by reputable companies, organizations, and institutions, there are some which seem to be run by marketing/PR agencies primarily for obtaining entry fees (the "better" contests have no fees or very nominal fees); for getting lots of IP at low/no cost; and for accruing attention and glamour for the contest sponsor rather than the entrants.
Have you ever voluntarily entered a design contest? What was your experience? When would you recommend someone decline, despite pressure to enter? When would you encourage participation?