Oh, rivalries exist. Most of the time it's between teams from the same city, especially if there's collaboration. You get students from occasionally rivalling schools together, and there's bound to be a bit of competition to see who can do better, not just against each other, but overall.
Karthik has a great speech about the big Canadian rivalries too (188 vs 2056/1114).
GTR 2010, F1.
My vote for most-storied rivalry? Gotta be 67 vs 1114:
2007. Curie Semis. 67 (1732+48) tops 1114 (469+1523) in 3 matches.
2008. Einstein Finals. 1114 (217+148) becomes World Champs (over 67+16+348).
2010. Einstein Finals. 67 (177+294) becomes World Champs (over 1114+469+2041).
2012. Archimedes Finals. 1114 (2056+4334) squeezes out a win vs 67 (2826+4143)
Fun fact: the last 2 times 67 met 1114 in the Champs Elims, they also had to beat 2056 (Einstein Semis 2010 (2056+1625+3138) and Archimedes Finals 2012) That's the birth of another great rivalry right there.