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anemone
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For the integers $a, b, c$, we have $\dfrac{2a-b}{c}=\dfrac{2b+c}{a}=\dfrac{-2a-c}{b}$.
If $a+b=2004$, find $a+b+c$.
If $a+b=2004$, find $a+b+c$.
anemone said:For the integers $a, b, c$, we have $\dfrac{2a-b}{c}=\dfrac{2b+c}{a}=\dfrac{-2a-c}{b}$.
If $a+b=2004$, find $a+b+c$.
$\dfrac{2a-b}{c}=\dfrac{2b+c}{a}$ | $\dfrac{2a-b}{c}=\dfrac{-2a-c}{b}$ | $\dfrac{2b+c}{a}=\dfrac{-2a-c}{b}$ |
$2a^2-ab=2bc+c^2$ | $(b+c)(2a-b+c)=0$ | |
But notice that $b+c\ne 0$ because if $b+c=0$, $2a^2-ab=2bc+c^2$ becomes $2a^2-ac=-2c^2+c^2$ $2a^2-ac+c^2=0$ $2a^2-ac+c^2=0$ $2(a-\frac{c}{4})+\frac{7c^2}{8}\ne 0$ for all integers $a, c$. Hence, $2a-b+c=0$ | $\dfrac{2b+c}{a}=\dfrac{-(2a+c)}{b}$ $\dfrac{2b+c}{a}=\dfrac{-b}{b}$ $2b+c=-a$ $a+2b+c=0$ |
let :anemone said:For the integers $a, b, c$, we have $\dfrac{2a-b}{c}=\dfrac{2b+c}{a}=\dfrac{-2a-c}{b}$.
If $a+b=2004$, find $a+b+c$.
The general method to solve this problem is to use algebraic equations and properties to manipulate the given information and ultimately solve for the value of a + b + c. This may involve substituting values, combining like terms, and using the fact that a + b = 2004.
Yes, it is possible to solve this problem without using algebra. One method could be to list out all the possible combinations of a, b, and c that add up to 2004 and then checking each combination to see if it satisfies the given conditions.
No, there are multiple solutions to this problem. Since there are three variables (a, b, and c) and only two equations (a + b = 2004 and a + b + c = ?), there will be infinitely many possible solutions.
The smallest possible value for a + b + c is 2004. This can be achieved when a and b are both 0 and c is 2004.
Yes, negative integers can be used in this problem. The only requirement is that a, b, and c are all integers. Therefore, the solution could involve positive, negative, or a combination of both types of integers.