In this paper we investigate how competition among tutees in the context of learning by teaching affects tutors' engagement as well as tutor learning. We conducted this investigation by incorporating a competitive Game Show feature into an online learning environment where students learn to solve algebraic equations by teaching a synthetic peer, called SimStudent. In the Game Show, pairs of SimStudents trained by students beforehand competed against each other by solving challenging problems to attain higher ratings. The results of a classroom study with 141 7th through 9th grade students showed the following: (1) Students improved their proficiency to solve equations after teaching SimStudent, but there was no observed improvement in their conceptual understanding. (2) Overall, the competitive Game Show promoted students' extrinsic and intrinsic motivations--when the competitive Game Show was available, students' engagement in tutoring (intrinsic motivation) was increased; students who arguably had a higher desire to win strategically selected opponents with lower proficiency for an easy win (extrinsic motivation). (3) The availability of the competitive Game Show did not affect tutor learning; there was no notable correlation between students' motivation (intrinsic or extrinsic) and tutor learning. Based on these findings, we propose design improvements to increase tutor learning.