A fun algebra question

Algebra Level 3

$a, b,$ and $c$ are real numbers such that $a+x^{2} = 2015$ , $b+x^{2}= 2016$ , $c+x^{2} = 2017$ and $a b c= 999$ . Find $\frac{a}{bc}+\frac{c}{ab}+\frac{b}{ac}-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}$

1

\frac{1}{666}

2016

\frac{1}{333}

0

1 solution

Rishabh Jain
Jun 16, 2016

Substract three eqns pairwise to get: $a-b=b-c=-1,a-c=-2$ .

Now write the given expression as:

$\dfrac{a^2+b^2+c^2-ab-bc-ca}{abc}$

$=\dfrac{(a-b)^2+(b-c)^2+(c-a)^2}{2abc}$ $=\dfrac{1+1+4}{2(999)}$

$\Large =\boxed{\dfrac1{333}}$

Aww man you beat me to it XD

Manuel Kahayon - 4 years, 12 months ago

This was simple And I'm 13

Owen Moriarty - 4 years, 12 months ago

A fun algebra question

1 solution

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