This letter is definitely an excellent start for your application to MIT. However, there are definitely some issues with your English grammar and it is good that you are seeking help.
For starters, in your first sentence, you have used the semicolon in an invalid way for English grammar. The semicolon can only be used to connect--to punctuate--two complete sentences that are so closely related that a period would be incorrect. It can also be used to string a complex list together, such as: Pismo Beach, California; Oceano, California; San Luis Obispo, California. In this example the list already requires commas, so we replace the comma standing in the place of "and" with a semicolon. You will want to rewrite your first sentence without the semicolon.
In your first paragraph you have also used the word competitive incorrectly. Competitive is an adjective which tells us more about or modifies nouns. In your phrase: "is so much competitive here," the word here is not a noun so competitive cannot be next to it. Instead you need to make competitive the noun form, because it is the object of your subject city. Instead, consider revising this second sentence to: "Since the city's population density is approximately 3000 people per square kilometer, there is a great deal of competition among the residents, regardless of whether or not they are competing for academic or non-academic purposes."
You also have some unclear phrases in your writing, such as, "But I long for achievements." It will be better English to change it to, "But I long to achieve," but then you would have to state what you want to achieve. You will also need to add a transitional phrase to better connect your quote, "Nothing to loose.." with the phrase "as I believe." Try something like, "as I believe there is 'nothing to loose....'"
You will also need to watch our for unclear antecedents. "This" is an unclear antecedent. The word "this" actually refers to what you said previously about your competitive background. Therefore you will want to substitute "this" for: “My competitive background makes me concerned about rivalry....” But then you next say that challenges and competitions are your pleasure, so we now see that your verb choice "concerned" is not the best choice. To be concerned means to be worried. If you are worried about rivalry than you will not also be taking pleasure in challenges and competitions. Instead of concerned, you may mean aware. Being aware means that you acknowledge that something exists. You may want to say that your background makes you aware and appreciative of rivalry in your academic and social life.
In your next paragraph, "Assistances around me" is also unclear English, namely because “assistances” is not a real English word. Instead, the English words are assistance and assistant. But furthermore, by this phrase, you could be referring to the people who have assisted you or you could be referring to your environment. Once you make it clear what assistance you are actually speaking of, you will want a clear transition between this idea and your science club. You may consider something like this for your second sentence: "One thing that helped create who I am today is my High School Science Club. I owe a lot to my science club."
There are other errors similar to these and correcting them will help.