We consider strategic problems in college admissions with score-limits introduced by Biró and Kiselgof. We first consider the problem of deciding whether a given applicant can cheat the algorithm of Biró and Kiselgof so that this applicant is assigned to a more preferable college. We prove its polynomial-time solvability. In addition, we consider the situation in which all applicants strategically behave. We prove that a Nash equilibrium always exists, and we can find one in polynomial time