If they see that you have not used the first sector, then at checkin they can insist on reissuing the ticket, and recalculating the fare. It is both expensive, and time consuming.

I recommend that you buy a one way, full fare to the starting point of your RTW. You can then follow it, and use the last sector to start your next one.

Out of LAX, it is realtively inexpensive to get yourself to YVR. You can then start a RTW here (paying the Canadian fare). Your last sector will be LAX-YVR, which you can use anytime in the next year to go and start your next one.

Depending on the savings involved, you can follow the same tactic to start tix in other cheap centres, like BKK.