A character in a story becomes romantically interested in someone and believes that the best way to get his/her attention is by being just as good as (s)he is, if not better, at something that (s)he does. The fields can be academic, athletic, or something more mundane like board game tourneys, as long as they can contend. The characters may be strangers to each other, or they may already be familiar with each other; either way, one person develops feelings for and decides to pursue the other. Neither person has to know about the other's harbored affections, or lack thereof, but the suitor must strongly feel that (s)he has something to prove in order to be taken seriously. Said suitor may start out as an Unknown Rival in the beginning, but since success depends on the individual's level of ambition, this is likely to be a plight characteristic of a determined, yet disciplined romantic. It can also happen in the reverse order, i.e 2 characters being straight rivals at first, then one of them coming to like the other's company and care about them. (Friendly Rivalry often also works like this, albeit platonically.) I Was Just Passing Through and The Only One Allowed to Defeat You may come from this.
The courter may actually enjoy the activity they're engaged in, going on to do great things with the prospective relationship giving him/her just one more reason to perform well. Or, alternatively, the courter may not really care for the activity (at least not initially) and drop it once the courtship ends, since the point of taking it up in the first place had always been to make a good impression on and successfully romance the other party, with whatever other achievements being corollary.
The type of significance one character will have to another will depend on the nature of their rivalry. At best, the object of affection will be left with the impression that they're into the same things and see the suitor as someone (s)he can banter and have fun with. Or, acknowledge the other party as a Worthy Opponent who will provide a challenge, as "iron sharpens iron". At worst, the Love Interest will hate the other person's guts (maybe because of bad sportsmanship from one side or the other), but that suitor will still prefer the negative attention to that love interest not even knowing (s)he exists.
Compare Friendly Rivalry or Defeat Means Friendship for non-romantic examples, although they could evolve into this trope. Contrast Foe Romantic Subtext and Dating Catwoman for when the two are outright enemies.
- A 2016 commercial for an upgraded Fitbit fitness tracker bracelet has a man watching a woman jog by. He tries to keep up with her but she outpaces him. After a montage of him working out, he tries again and does well enough to gain her notice... before she hits the next leg of her route: a flight of stairs built up a tall hill. Then he is left behind once again. The ad was soon removed because although it was intended to look romantic, the alternate interpretation was that the guy was a bit of a creep for pursuing his interest this way.
- Aoi and Sakura's romance in Love Gene XX grows out of their rivalry (mostly one-sided, as Aoi is the only one obsessed with it) over the position of the top student of the Kingdom Academy.
- Mindy Crenshaw from Drake & Josh is introduced in the first season as a classmate who outdoes Josh, (specifically in the sciences), believes herself to be smarter than him and insults his intelligence every chance she gets. It's not until the season 3 episode "Mindy's Back" after Josh gets fed up and confronts her that Mindy confesses that all her arrogance was a front to hide her admiration for him. From that point on, they end up in an on-and-off relationship until the Series Finale.
- The Relationship Values in Dragon Age II are placed on the Friendship-Rivalry scale, depending on whether the respective party member sees Hawke as a confidante to trust or as a contrarian to prove wrong. Initiating and completing a Romance Sidequest with any of the four Love Interests is possible regardless of whether you are friends or rivals with them (only the overall intensity of the relationship counts), and romancing a rival is very much a Slap-Slap-Kiss deal.
- The Adventures of Jimmy Neutron: Boy Genius has both Cindy Vortex and Jimmy himself, who both even have their own respective crushes (Nick Dean and Timmy Turner for Cindy, Betty Quinlan and "Win, Lose and Kaboom's" April for Jimmy) while maintaining their own academic rivalry, competing for the reputation of "smartest kid in school". Over the course of the show, however, their relationship develops and goes onto encompass instances of mutual jealousy, an aphrodisiac ("Love Potion 976/J"), and a future timeline where they are expressly married.