What kind of glues can you use? Can you use caulk on the seams? How about paint? You have two major challenges: fastening the panels along the seams and sealing the cardboard so that water doesn't destroy it instantly. As for plans, I would take this $10 plan:
http://www.bateau.com/proddetail.php?prod=NC16 and modify it for construction in cardboard. The great thing about this pirogue is that it is meant to be built with plywood, which is another material which only bends in one plane. This plan is designed around conical sections, which are easily formed with plywood AND corrugated board. Let us know if there are more specific rules.
(I like this 15' 9" design for your race because it will easily carry two people, and be easy to paddle. The same designer offers free plans for the same boat in a 13'9" length, but it would only hold two small paddlers:
http://www.bateau2.com/free/cheapcanoe.htm.)
For what it's worth, I took a Boy Scout troop and built seven of the NC-16s. It's an easy boat to build, and with some clever modifications, would work well in cardboard. I'm tempted to build one just for fun...