The phrase "within the competition" is correct and usable in written English.
You can use it to refer to activities, challenges, events, etc. that are part of a larger competition or contest. For example: "The athletes had to compete in three events within the competition to win the championship title."
"You have a nice friendly competition within the competition of the team," Randolph said, "and I think good staffs feed off that type of energy.
This book was an amazing read and full of adventure and anticipation as to whether she would be successful within the competition.
In what proved to be a competition within the competition, the four Americans who did swim were essentially racing one another for the privilege to team with Lezak, Weber-Gale and Phelps.
In what proved to be a competition within the competition, the four Americans who did swim were essentially racing one another for the opportunity to team with Lezak, Weber-Gale and Phelps in the final.
He always wanted to do a little bit better, and that's how they stayed so sharp over the years, by always competing — obviously to beat the other team, but also the competition within the competition".
In addition, the percentile ranks of 159 students will rise, within the competition for districtwide or citywide programs, though it was unclear what effect that would have on winning a seat.
Competition within the competitions can be fierce, however.
Then, it is preferable to internalize the effects of competition within substitutes, the competition effect, by partially monopolizing one component type.
As more smartphone makers include IR ports within their devices, the competition will get tougher.
Within the tablet market the competition is fierce.
The competition within the street is going to diminish a lot over time.
