Have you considered sponsoring engineering email newsletters? They have readers interested in engineering and getting into their inbox means you're much more likely to be noticed than if you were just some banner on the side of a website. LaunchBit is an email ad network that can totally help you.
Put fliers at engineering schools.. if you put that up at Purdue campuses for example you'd get groups of students working together for big money...
actually if you had $1000000 you'd advertise it like the X-prize.
Spend $500,000 on an audacious advertising budget with a top agency and lower the prize money.
One does not simply explain such a big thing in one reply on Yahoo! Answers.
I want to know how to promote a competition that awards $1,000,000 to anyone that can solve an engineering problem. How can I promote and market this?