verb

strive to gain or win something by defeating or establishing superiority over others.

universities are competing for applicants

take part,play,be a contestant,be a competitor,participate,be involved,get involved,engage,enter,go in for,throw one’s hat in the ring,be in the running,contend,vie,fight,battle,clash,tussle,grapple,wrestle,wrangle,jockey,wage war,cross swords,lock horns,go head to head,strive against,struggle against,pit oneself against,challenge,take on,try to beat,pitch oneself against