I think your looking for
Destined for Me by: cindygirl.
Sesshomaru was Lord of the Western Lands. He had been ruling for 224 years. But a law states that he must find a mate before he reaches 225 of being lord to the land. An old medalleon given to him by his fathers best friend will help him seek his mate.
Rated: MA
www.fanfiction.net/s/2632933/1/Destined-for-Me
It has a want-to-be-mate in the picture. But sesshomaru want nothing to do with her. The mate test has two parts (intellectual capability and battle royal) and she is tested by the three of the youkai lords. The dragon comes into the story at the end of the test.