You're both going to have to learn to control your own egos in this particular pairing. Even if you're ordinarily not the competitive type, something will happen when the two of you get together that accentuates your egos and your drives for success. In fact, success and achievement will take on too much importance over time, if you don't do anything to temper the situation.
Other more harmonizing influences in your relationship will help, but in general, you'll both find it hard to support each other -- as if your partner's successes can only happen at the expense of your own. Don't allow things to degenerate into a petty struggle for power or domination over each other; neither one of you will win that battle. Instead, focus on checking your own ambitious and competitive urges, and on supporting your partner no matter how uncomfortable it makes you feel. Perhaps you'll even find a way to combine your egos and energies and work together toward a common goal, one that benefits not only yourselves but the world around you.