Looking on the internet deeply has found these results:

compete with each other is the most popular phrase on the web.

compete among other 10 results on the web	compete with each other 1304,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
It doesn't look like this phrase is very popular! The search could not find any examples on the internet.	intr.v. com�pet�ed, com�pet�ing, com�petes. To strive against another or others to attain a goal, such as an advantage or victory. See Synonyms at rival. Girls Compete With Each Other, Women Empower One Another. Posted: 10/21/ 2014 4:29 pm EDT Updated: 12/21/2014 5:59 am EST. TWO WOMEN LOOKING �... Flowers also compete with each other without stepping on each other, do you think human beings will achieve this healthy level of competition ... At IIM A, do�... Two or more persons can compete with each other as teammates, however compete with is also used to indicate two persons competing against each other.

compete among other

10 results on the web

compete with each other

1304,000 results on the web

More popular!

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

intr.v. com�pet�ed, com�pet�ing, com�petes. To strive against another or others to attain a goal, such as an advantage or victory. See Synonyms at rival.
Girls Compete With Each Other, Women Empower One Another. Posted: 10/21/ 2014 4:29 pm EDT Updated: 12/21/2014 5:59 am EST. TWO WOMEN LOOKING �...
Flowers also compete with each other without stepping on each other, do you think human beings will achieve this healthy level of competition ... At IIM A, do�...
Two or more persons can compete with each other as teammates, however compete with is also used to indicate two persons competing against each other.