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Allah (SWT) says in the Quran: 
If you are in doubt about the revelation We have sent down to Our servant, then produce a single chapter like it, and call upon your helpers besides God, if you are truthful. 
But if you cannot do it, and you can never do it,…
(Qur'an    2 : 23 - 24)
Say: "If the whole of mankind and Jinns were to gather together to produce the like of this Qur'an, they could not produce the like thereof, even if they backed up each other with help and support." (Qur'an 17 : 88)
Ever since the Qur'an was revealed, fourteen centuries ago, no one has been able to produce a single chapter like the chapters of the Qur'an in their beauty, eloquence, splendor, wise legislation, true information, true prophecy, and other perfect attributes. 
Note of this point, 
the smallest chapter in the Quran, chapter 108, has only ten words. Yet no one has ever been able to meet this challenge, then or today. 
 
Some of the disbelieving arabs who were enemies of the Prophet Muhammad 'pbuh' tried to meet this challenge to prove that Muhammad 'pbuh' was not a true Prophet, but they failed to do so. This failure was despite the fact that the Qur'an was revealed in their own language and dialect and that the arabs at the time of Muhammad 'pbuh' were  very eloquent people who used to compose beautiful and excellent poetry that is still read and appreciated today. 
Allah (SWT) says in the Quran: "Do they not ponder on the Qur'an (with care)? Had it been from other than Allah, they would surely have found therein much discrepancy." (Qur'an : 4 : 82)
 
  
     
     
     
    
    
those who shall follow the [last] Apostle, the unlettered Prophet... (Quran 7:157)
You were not able to read or write before the Quran was revealed to you; otherwise, the followers of falsehood would have tried to confuse the matter.  (Quran 29:48)
 
  
     
     
     
    
    
It is true that Prophet Muhammed [PBUH] did not read or write.
Whenever the revelation came down to Prophet Mohammed, he memorized the verses regardless of how long of how many they were from the first time and then dictated them to his companions, who wrote them down immediately. Hence, the Quran was both memorized and written in different manuscripts at the same time.
Sometimes, when Prophet Mohammed Peace Be Upon Him received the revelation, he used to repeat the verses immediately, fearing that he may forget the revealed verses. So Allah Almighty ordered the Prophet to follow the recitation of the angel:
(O Prophet,) do not move your tongue (during revelation) for (reciting) it (the Qur‘an) to receive it in hurry 
It is surely undertaken by Us to store it (in your heart), and to let it be recited (by you after revelation is completed) 
Therefore, when it is recited by Us (through the angel), follow its recitation (by concentration of your heart)
Then, it is undertaken by Us to explain it. 
[Quran 75 : 16 - 19]
After the death of Prophet Mohammed Peace Be Upon Him, the first Caliph Abu Bakr Al Siddeeq appointed the companion Zayd Bin Thabit to lead the task of compiling the Quran. Zayd collected the various manuscripts and made sure that the new compilation includes the chapters that are both written in the manuscripts and memorized by hearts.
With the expansion of Islam into new territories, people started reciting with the different dialects approved by the Prophet, but every party thought that only his dialect was correct. Caliph Othman was made aware of this. Accordingly, Othman formed a committee of known and trusted memorizers (this was almost 14 years after the death of Prophet Mohammed). The committee included 5–12 members, including Zayd Bin Thabit. The committee made copies of the Quran, based on the original copy generated by Zayd Bin Thabit at the time of Abu Bakr Al-Siddeeq. Historians believe that the committee made 7 copies of the Quran, which were sent to Mecca, Al-Koofa and Al-Basrah, Al-Shaam (Levant), Yemen, Bahrain, and one copy was kept in Madinah.
Allah SWT said in the Quran : 
Surely We have revealed this reminder (The Qur'an); and We will surely preserve it Ourself. 
(Quran 15:9)
 
  
     
     
     
    
    
Narrated Abu Huraira:
The Prophet (ﷺ) said, "There was no prophet among the prophets but was given miracles because of which people had security or had belief, but what I was given was the Divine Inspiration which Allah revealed to me. So I hope that my followers will be more than those of any other prophet on the Day of Resurrection."  
[Sahih al-Bukhari 7274]
...No Messenger was to show miracles without the permission of God.
(Quran 13:38)
Allah SWT supported His Prophets with miracles, which are extraordinary events that are usually of the same nature that the people of that particular prophet excelled at.  
For instance, the people of Prophet Moosa (AS) excelled at magic, so the miracle of Prophet Moosa (AS) was that he threw his staff and it turned to a serpent.  Likewise, the people of Prophet Eesa (AS) excelled at medicine, so Prophet Eesa (AS) cured the blind and the leper and gave life to the dead by the permission of Allah SWT.  
Similarly, the Arabs excelled at poetry and rhetoric and therefore Allah SWT sent them the Quran and it was a miracle that they were not even able to compose one Soorah (chapter) like it.  Qur'an also contains miracles in various fields including science, which have been proving to be accurate as they are being discovered.