In a team competition, two participating teams have an equal number of players, and each team orders its players linearly based on their strengths. A mechanism then specifies how the players from the two teams are matched up and how to score them. There are two types of manipulations by a team: Misreporting the strength ordering and deliberately losing a match. To identify these strategically behaviors, we model the team competition problem in a game-theoretical framework, under which we prove necessary and sufficient conditions which ensure that truthful reporting and maximal effort in matches are equilibrium strategies, and which further ensure certain fairness conditions described by choice functions. Categories and Subject Descriptors J.4 [Social and Behavior Sciences]: Economics; I.2.11 [Distributed Artificial Intelligence]: Multi-agent System General Terms Economics Keywords Team Competition, Mechanism Design, Truthfulness, Dominant Strategy Implementation