These are mostly jealousy fics, but there is some rivalry:
Unwed Wizard: Capture the Dragon by fantasyfiend09 (http://serpentinelion.livejournal.com/371266.html) (really good)
For As Long As You Want by pessen03 (http://hd-smoochfest.livejournal.com/73530.html)
Envy by yeaka (http://highstakes-ww.livejournal.com/24443.html) (very intense but hot: read all warnings!)
Also, enchanted_jae (http://enchanted-jae.livejournal.com/) has a jealousy tag that has a lot of cute little rivalry fics, like Schooling the Instructor (http://enchanted-jae.livejournal.com/1731263.html#cutid1), Why Nott? (http://enchanted-jae.livejournal.com/1713656.html#cutid1), & A Date with Draco Malfoy (http://enchanted-jae.livejournal.com/979052.html#cutid1).
I'm not sure if these exactly fit your prompt, but I hope they work! <:)
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timeasmymeasure
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Date: 2013-07-26 04:59 pm (UTC)Unwed Wizard: Capture the Dragon by fantasyfiend09 (http://serpentinelion.livejournal.com/371266.html) (really good)
For As Long As You Want by pessen03 (http://hd-smoochfest.livejournal.com/73530.html)
Envy by yeaka (http://highstakes-ww.livejournal.com/24443.html) (very intense but hot: read all warnings!)
Also, enchanted_jae (http://enchanted-jae.livejournal.com/) has a jealousy tag that has a lot of cute little rivalry fics, like Schooling the Instructor (http://enchanted-jae.livejournal.com/1731263.html#cutid1), Why Nott? (http://enchanted-jae.livejournal.com/1713656.html#cutid1), & A Date with Draco Malfoy (http://enchanted-jae.livejournal.com/979052.html#cutid1).
I'm not sure if these exactly fit your prompt, but I hope they work! <:)