Question

If $a,b,c$ are three distinct positive real numbers such that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}>k$, then the greatest value of $k$ is:

• A. $9$
• B. $3$
• C. $4$
• D. $6$

Answer 1

Answer: (D)

$6$

Given, $a,b,c\in +R⟹a\ge 0,b\ge 0,c\ge 0$

But $a,b,c$ cannot be equal to zero as they are the denominators

Therefore, $a\ge 1,b\ge 1,c\ge 1$

$⟹\frac{b}{a}\ge 1,\frac{b}{c}\ge 1,\frac{c}{a}\ge 1,$

$\frac{a}{b}\ge 1,\frac{c}{b}\ge 1,\frac{a}{c}\ge 1$

adding all these we get

$⟹\frac{b}{a}+\frac{c}{a}+\frac{a}{b}+\frac{c}{b}+\frac{a}{c}+\frac{b}{c}\ge 1+1+1+1+1+1$

$⟹\frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}\ge 6$

Therefore, the greatest value of $k$ is $6$