Let $a,\,b$ and $c$ be real numbers such that $(a-b)^3+(b-c)^3+(c-a)^3=9$. Prove that $\displaystyle \frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}\ge \sqrt[3]{3}$.

Let $a,\,b$ and $c$ be real numbers such that $(a-b)^3+(b-c)^3+(c-a)^3=9$.

Prove that $\displaystyle \frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}\ge \sqrt[3]{3}$.

My solution:

I know there are students, who upon seeing the equality wanted to expand the cube inside the parentheses and get started from there.

I won't say that is not a good start to prove the intended inequality, but, if we could, we should always look for ways to simplify things, rather than to complicate the problem or given data/information at hand.

Therefore, our next effort should be channeled into simplifying the information that we're given:

$(a-b)^3+(b-c)^3+(c-a)^3=9$

When the right situation arises, we could always use the help of a substitution.

Let $x=a-b,\,y=b-c$, then if we add up $x$ and $y$, we get another relation that says:

$a-b+b-c=a-c=x+y\implies\,\,\,-(x+y)=c-a$

Therefore $(a-b)^3+(b-c)^3+(c-a)^3=9$ becomes:

$x^3+y^3+(-(x+y))^3=9$

$x^3+y^3-x^3-3xy(x+y)-y^3=9$

$xy(x+y)=-3$

$\displaystyle \frac{1}{x+y}=-\frac{xy}{3}---(1)$

Also, the intended inequality becomes

$\displaystyle \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{(-(x+y))^2}\ge \sqrt[3]{3}$

$\displaystyle \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{(x+y)^2}\ge \sqrt[3]{3}---(2)$

If we replace $\displaystyle \frac{1}{x+y}=-\frac{xy}{3}---(1)$ into the LHS of the expression in the inequality (2), we see that we have:

$\displaystyle \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{(x+y)^2}$

$\displaystyle =\frac{1}{x^2}+\frac{1}{y^2}+\frac{x^2y^2}{9}$

$\displaystyle \ge 3\left(\frac{1}{x^2} \cdot \frac{1}{y^2} \cdot \frac{x^2y^2}{9}\right)^{\frac{1}{3}}$ (by the AM-GM inequality)

$\displaystyle \ge 3\left(\frac{1}{3^2}\right)^{\frac{1}{3}}$

$\displaystyle \ge 3^{\frac{1}{3}}$

and we're hence done.

I will do another post for expanding the given equality to see where that leads us, will it lead us to the simplicity or complexity? I bet you have already an answer for it...