This quote doesn't seem to make grammatical sense, did you mean to say teammates in the plural or singular?
As I discuss the ways to solve the problem with my teammates, Edward, who reminds me of suggests in a trembling voice that we solve the problem using trial and error, but I know that there is a better way to solve the problem.
I like this whole sentence but you could change this:
and we are dreading to hear the bell that signals the end of the round since we have spent so much time on this question
to: and we are dreading hearing the bell that signals the end of the round, since we have spent so much time on this question.
As I sort through the problem using the solution method that my teammate who reminds me of a character from Mean Girls, my senses are suddenly enhanced: I can hear the conversations of other teams, feel the hexagonal ridges of my yellow pencil, and smell the sweat of me and my teammates. This problem caused a crisis for the team, and the general outlook on the competition of team members was bleak.
This first part (As I...Mean Girls) doesn't make sense to me. Also change "smell the sweat of me and my teammates" to "smell the sweat of my teammates and I".
Your last paragraph is really good, but I feel like you've used "problem" a lot...maybe look in a thesaurus and see if there are any other words that could fit the meaning you're portraying.
Overall it's really good and shows your passion, you come across well in this essay!