Błędnie: Though the two boys used to conquer with each other when they were at college, today Peter is helping Paul a lot in his career. Poprawnie: Though the two boys used to compete with each other when they were at college, today Peter is helping Paul a lot in his career.
"Conquer" to 'podbić, zdobywac', 'zwyciężać, pokonywać'. 'Konkurować, rywalizować'to po angielsku "compete (with)".