Thanks very much for confirming my input as your answer.
I'll repost it here for the sake of future readers.
I'm not sure there are really many workable alternatives for this.
If speed is an issue in the competition, the ultimate speed involves
simply going hand over hand, if your arms are powerful to allow you
to use this method.
"The world record of 2.8 seconds for the 20' climb established by
Don Perry of UCLA is faster than most people can pull a loose 20'
rope on the floor between their legs!"
Short of that, about the best and only option I know of is to use
both hands to pull yourself up, then lift your knees and grab the
rope between your locked legs and pull yourself up to a higher
hand hold, repeating the process.
About the only variation of that I'm aware of is in where between
your legs you grab the rope. If you're wearing shoes with a good
grip, you may be able to emulate the coconut tree climbers in the
islands, and grab the rope between your feet, which would allow
for a higher raise when you push up, however, in practice, I
think you'll find this impractical and too fatiguing to continue
for 18 feet. In that case you'll settle for the way most people
do it, which is to lock the rope between your legs somewhere in
the vicinity of your knees, or, if possible, somewhere between
your knees and ankles, but this can more awkward than using feet.
You'll see what I mean when you try it.
A lot will depend on the rules for the competition, but
from the way you phrased the question, it sounds like
you intend to (and will be allowed to) use your legs
during the climb, which is definitely in keeping with
your goal of making the climb as easy as possible.
Additional information may be found from an exploration of
the links resulting from the Google searches outlined below.
Searches done, via Google:
"rope climbing techniques"
"rope climbing techniques" -top
"rope climbing techniques" -top -mountain -wall
gymnastic "rope climbing techniques" -top -mountain -wall