If you surmount a challenge or difficult situation, you're not just getting over it. You're outdoing yourself, exceeding expectations, and overcoming the task at hand.
This word comes to us from the Old French verb for mount, or climb. Maria sang "climb every mountain..." in The Sound of Music. But she could have just as well meant "surmount every obstacle" — because to surmount means to both reach the highest point of something, like a mountain, and to totally overcome a mountain-size problem.