Hikari does win one competition that she sees herself in with Kei - the first to fall in love loses. But because she never realizes that he fell for her first, even after they do get together, she misses that she wins the biggest competition between the two.
Despite being the principle plot of most of the series, Hikari actually technically beats Kei several times through the run of things without it being brought up.
She wants to beat him in an OFFICIAL contest. Those technicalities wouldn't count.
