Asked by Allen
                Can anyone help with this question? 
Your backpack contains 2010 chocolate bars, which you are going to divide between your best friend and yourself. You think you are a nice person, so you will give your best friend more than zero chocolate bars. However, you have been craving chocolate recently, and so you must have at least twice as many chocolate bars as your friend after you finish dividing them. What is the difference between the maximum and minimum number of chocolate bars you can have?
            
        Your backpack contains 2010 chocolate bars, which you are going to divide between your best friend and yourself. You think you are a nice person, so you will give your best friend more than zero chocolate bars. However, you have been craving chocolate recently, and so you must have at least twice as many chocolate bars as your friend after you finish dividing them. What is the difference between the maximum and minimum number of chocolate bars you can have?
Answers
                    Answered by
            Steve
            
    let's say you have exactly twice as many bars as he. So you have 1340 and he has 670
So, as long as you have at least 1340, but fewer than 2009 bars, he has some, and you have at least twice as many.
    
So, as long as you have at least 1340, but fewer than 2009 bars, he has some, and you have at least twice as many.
                    Answered by
            Bob
            
    Sorry, but it's wrong...
    
                    Answered by
            Anonymous
            
    669
    
                    Answered by
            324234
            
    it is 670
    
                    Answered by
            Kaitlyn
            
    it is actually 669  
    
                    Answered by
            669
            
    The answer is my name
    
                    Answered by
            Leon Zhang
            
    I bet you got it from AoPS
    
                    Answered by
            Anonymous
            
    i did
    
                    Answered by
            The answer is 669
            
    669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 669 
    
                    Answered by
            The answer is 669
            
    Since your friend receives at least $1$ chocolate bar, the maximum number of chocolate bars you can get is $2010 - 1 = 2009$. If you receive $n$ chocolate bars, then your friend gets $2010-n$ chocolate bars. Since you must have at least twice as many chocolate bars as your friend, we have the inequality\begin{align*}
n &\ge 2(2010-n) \\
\Rightarrow\qquad n &\ge 4020-2n \\
\Rightarrow\qquad 3n &\ge 4020 \\
\Rightarrow\qquad n &\ge 1340.
\end{align*}Therefore, you must keep a minimum of $1340$ chocolate bars. Therefore, the difference between the maximum and minimum number of chocolate bars you can have is $2009 - 1340 = \boxed{669}$.
    
n &\ge 2(2010-n) \\
\Rightarrow\qquad n &\ge 4020-2n \\
\Rightarrow\qquad 3n &\ge 4020 \\
\Rightarrow\qquad n &\ge 1340.
\end{align*}Therefore, you must keep a minimum of $1340$ chocolate bars. Therefore, the difference between the maximum and minimum number of chocolate bars you can have is $2009 - 1340 = \boxed{669}$.
                    Answered by
            Responder
            
    I got this from AoPs and thats why i searched it up
    
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