Tricky abc

Algebra Level 3

Let $a$ , $b$ and $c$ be distinct real numbers such that

$\large a+\frac{2}{b}=b+\frac{2}{c}=c+\frac{2}{a}$

Find the value of $|abc|$ .

Hint: You don't need to find the exact values of $a$ , $b$ and $c$ . Try to find the answer in another way.

4 2√2 √2 8

1 solution

Ong Zi Qian
Oct 20, 2017

Due to $a+\frac{2}{b}=b+\frac{2}{c}$ ,

$a-b=\frac{2}{c}-\frac{2}{b}$

$a-b=\frac{2(b-c)}{bc}$

$bc=\frac{2(b-c)}{a-b}$ ---------------------------------(*)

With the same way, we get

$b+\frac{2}{c}=c+\frac{2}{a}$ ,

$b-c=\frac{2}{a}-\frac{2}{c}$

$b-c=\frac{2(c-a)}{ac}$

$ac=\frac{2(c-a)}{b-c}$ ---------------------------------(*)

$c+\frac{2}{a}=a+\frac{2}{b}$ ,

$c-a=\frac{2}{b}-\frac{2}{a}$

$c-a=\frac{2(a-b)}{ab}$

$ab=\frac{2(a-b)}{c-a}$ ---------------------------------(*)

Multiply the three (*) equations, we get

$bc \cdot ac \cdot ab=\frac{2(b-c)}{a-b} \cdot \frac{2(c-a)}{b-c} \cdot \frac{2(a-b)}{c-a}$

$(abc)^2=8$

$|abc|=\boxed{2\sqrt2}$

Nice solution, but to be complete, can you give an example of 3 distinct positive real numbers $a,b,c$ that satisfy the original system of equations?

Brian Charlesworth - 3 years, 7 months ago

Your suggestion is good. I will did my best to try it. Thank you.

Ong Zi Qian - 3 years, 7 months ago