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Let $a,b,c$ be positive real numbers such that $\dfrac{1}{bc}+ \dfrac{1}{ca}+ \dfrac{1}{ab} \ge 1$. Prove that $\dfrac{a}{bc}+ \dfrac{b}{ca}+ \dfrac{c}{ab} \ge 1$.

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4 Answers 4

up vote 3 down vote accepted

Denote $X=\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab}$ and $Y=\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}$. In fact, if $X\ge 1$, then $Y\ge \sqrt{3}$. ($\sqrt{3}$ is optimal, because when $a=b=c=\frac{1}{\sqrt{3}}$, $X=1$ and $Y=\sqrt{3}$.)

By Cauchy-Schwarz inequality,

$$3(a^2+b^2+c^2)\ge (a+b+c)^2.$$

It follows that

$$Y=\frac{a^2+b^2+c^2}{abc}\ge \frac{(a+b+c)^2}{3abc}=\frac{abc}{3} X^2\ge \frac{abc}{3}.$$

If $abc\ge 3\sqrt{3}$, we are done. Otherwise, by inequality of arithmetic and geometric means,

$$Y\ge 3(abc)^{-\frac{1}{3}}\ge \sqrt{3}.$$

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Remember, that for any nonzero $x$, one has

$$ x+\frac{1}{x} \geq 2 \tag{1} $$

We apply this inequality repeatedly. Put

$$ T=\dfrac{a}{bc}+ \dfrac{b}{ca}+ \dfrac{c}{ab} $$

We have $$ T=\frac{a^2+b^2}{abc}+\frac{c}{ab} =\frac{\sqrt{a^2+b^2}}{ab} \bigg(\frac{\sqrt{a^2+b^2}}{c}+\frac{c}{\sqrt{a^2+b^2} }\bigg) \geq \frac{2(\sqrt{a^2+b^2})}{ab}, $$

by using (1) with $x=\frac{\sqrt{a^2+b^2}}{c}$. Then,

$$ T \geq \frac{2}{\sqrt{ab}} \sqrt{\frac{a}{b}+\frac{b}{a}} \geq \frac{2\sqrt{2}}{\sqrt{ab}} $$ by using (1) with $x=\frac{a}{b}$. So $T \geq \sqrt{\frac{8}{ab}}$.

We see that if $ab \leq 8$, we are done. So we may assume $ab \geq 8$. By symmetry, we may also assume $ac \geq 8, bc \geq 8$. But then

$$ 1 \leq \dfrac{1}{bc}+ \dfrac{1}{ca}+ \dfrac{1}{ab} \leq \frac{1}{8}+\frac{1}{8}+\frac{1}{8} =\frac{3}{8}, $$

which is impossible.

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I think you forgot a $\sqrt$ in the line after the "We have", in the numerator of the first factor in the third expression from the left. –  user3533 Jan 19 '13 at 14:00
    
@user3533 : Fixed, thanks. –  Ewan Delanoy Jan 19 '13 at 14:02
    
Very good answer. I don't understand the downvote. –  user3533 Jan 19 '13 at 14:06
    
How do you justify the fact that assuming $ab\geq 8$ allows to assume $bc\geq 8$ and $ca\geq 8$? If it's boring to type, I would appreciate onilne links instead. P.S. it was not me who downvoted. –  007resu Jan 19 '13 at 14:09
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@Freddy: If you take the proof that $ab \leq 8$ is enough and exchange $a$ and $c$ whenever they appear you will get a proof that $ca \leq 8$ is enough. –  user3533 Jan 19 '13 at 14:15

I want to give a complete answer.

Assume by the sake of contradiction that $$\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}<1.\tag{1}$$ By Cauchy Schwarz it follows from $(1)$ that $$\frac{3}{abc}>\frac{3}{abc}\left(\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}\right)\geq \left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)^2.\tag{2}$$

From $(2)$ we recover two facts

  • By $AM-GM$ mean $$\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)^2\geq 3\left(\frac{1}{a^2bc}+\frac{1}{ab^2c}+\frac{1}{abc^2}\right),$$ therefore, combined with $(2)$, we obtain on one hand $$1>\frac{1}{a}+\frac{1}{b}+\frac{1}{c},\tag{3}$$ which implies that $\min\{a,b,c\}>1$ (remember that they are all positive).
  • On the other hand we can also derive from $(2)$ and $(3)$ the following: $$\frac{1}{\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)}>\frac{a+b+c}{3}>1,\tag{4}$$

which in the end leads to an absurd with respect to the hypothesis of the problem.

Then $(1)$ must be false and the problem is solved.

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This is my solution.

First we prove that $a^4+b^4+c^4\geq abc(a+b+c)\tag{1}$ We can get $a^2+b^2+c^2\geq ab+bc+ac$ easily by such $x^2+y^2\geq 2xy$.

Therefore: $$a^4+b^4+c^4\geq a^2b^2+b^2c^2+a^2c^2=(ab)^2+(bc)^2+(ac)^2\geq (ab)(bc)+(bc)(ac)+(ac)(ab)=abc(a+b+c)$$

We have $$(a^2+b^2+c^2)^2=a^4+b^4+c^4+a^2(b^2+c^2)+b^2(a^2+c^2)+c^2(a^2+b^2)$$ $$a^2+b^2\geq 2ab$$ $$a^2+c^2\geq 2ac$$ $$b^2+c^2\geq 2bc$$

and $(1)$, so $$(a^2+b^2+c^2)^2\geq abc(a+b+c)+2a^2bc+2b^2ac+2c^2ab=3abc(a+b+c)$$

Since $$\dfrac{1}{bc}+ \dfrac{1}{ca}+ \dfrac{1}{ab} \geq 1$$ We have $$a+b+c\geq abc$$ so $$(a^2+b^2+c^2)^2\geq 3abc(a+b+c)\geq 3(abc)^2$$ Therefore: $$a^2+b^2+c^2\geq \sqrt{3}abc> abc$$ Finally we get $$\dfrac{a}{bc}+ \dfrac{b}{ca}+ \dfrac{c}{ab} \ge 1$$ In fact $$\dfrac{a}{bc}+ \dfrac{b}{ca}+ \dfrac{c}{ab} \ge \sqrt{3} > 1$$

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