Ann and Cindy became rival since their first day at school.Wanting to sit on same chair had started it.All through its school years they fought for be number one in academic work and sports.At the Christmas party they compete to win the prize for the best cake.
第一句rival应该是rivals(对手都是成对出现的,所以要用复数哦).
第二句的那个,sit on the same chair,（同一张椅子,the same,same在表示同一个的时候,都是the same这样出现的）
第三句fought for后边应该接sth,所以那个be直接删除就是了.
第四句,时态有问题——上文全部都是过去时,所以那个compete变成competed就是了.