IMO Shortlist, 2006

We first consider the cubic polynomial
$$P(t)=t b\left(t^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c t\left(c^{2}-t^{2}\right)$$
It is easy to check that $P(b)=P(c)=P(-b-c)=0$, and therefore
$$P(t)=(b-c)(t-b)(t-c)(t+b+c)$$
since the cubic coefficient is $b-c$. The left-hand side of the proposed inequality can therefore be written in the form
$$\left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right|=|P(a)|=|(b-c)(a-b)(a-c)(a+b+c)|$$
The problem comes down to finding the smallest number $M$ that satisfies the inequality
$$\begin{equation*} |(b-c)(a-b)(a-c)(a+b+c)| \leq M \cdot\left(a^{2}+b^{2}+c^{2}\right)^{2} \tag{1} \end{equation*}$$
Note that this expression is symmetric, and we can therefore assume $a \leq b \leq c$ without loss of generality. With this assumption,
$$\begin{equation*} |(a-b)(b-c)|=(b-a)(c-b) \leq\left(\frac{(b-a)+(c-b)}{2}\right)^{2}=\frac{(c-a)^{2}}{4} \tag{2} \end{equation*}$$
with equality if and only if $b-a=c-b$, i.e. $2 b=a+c$. Also
$$\left(\frac{(c-b)+(b-a)}{2}\right)^{2} \leq \frac{(c-b)^{2}+(b-a)^{2}}{2}$$
or equivalently,
$$\begin{equation*} 3(c-a)^{2} \leq 2 \cdot\left[(b-a)^{2}+(c-b)^{2}+(c-a)^{2}\right] \tag{3} \end{equation*}$$
again with equality only for $2 b=a+c$. From (2) and (3) we get
$$\begin{aligned} & |(b-c)(a-b)(a-c)(a+b+c)| \\ \leq & \frac{1}{4} \cdot\left|(c-a)^{3}(a+b+c)\right| \\ = & \frac{1}{4} \cdot \sqrt{(c-a)^{6}(a+b+c)^{2}} \\ \leq & \frac{1}{4} \cdot \sqrt{\left(\frac{2 \cdot\left[(b-a)^{2}+(c-b)^{2}+(c-a)^{2}\right]}{3}\right)^{3} \cdot(a+b+c)^{2}} \\ = & \frac{\sqrt{2}}{2} \cdot\left(\sqrt[4]{\left(\frac{(b-a)^{2}+(c-b)^{2}+(c-a)^{2}}{3}\right)^{3} \cdot(a+b+c)^{2}}\right)^{2} \end{aligned}$$
By the weighted AM-GM inequality this estimate continues as follows:
$$\begin{aligned} & |(b-c)(a-b)(a-c)(a+b+c)| \\ \leq & \frac{\sqrt{2}}{2} \cdot\left(\frac{(b-a)^{2}+(c-b)^{2}+(c-a)^{2}+(a+b+c)^{2}}{4}\right)^{2} \\ = & \frac{9 \sqrt{2}}{32} \cdot\left(a^{2}+b^{2}+c^{2}\right)^{2} . \end{aligned}$$
We see that the inequality (1) is satisfied for $M=\frac{9}{32} \sqrt{2}$, with equality if and only if $2 b=a+c$ and
$$\frac{(b-a)^{2}+(c-b)^{2}+(c-a)^{2}}{3}=(a+b+c)^{2} .$$
Plugging $b=(a+c) / 2$ into the last equation, we bring it to the equivalent form
$$2(c-a)^{2}=9(a+c)^{2} .$$
The conditions for equality can now be restated as
$$2 b=a+c \quad \text{ and } \quad(c-a)^{2}=18 b^{2} .$$
Setting $b=1$ yields $a=1-\frac{3}{2} \sqrt{2}$ and $c=1+\frac{3}{2} \sqrt{2}$. We see that $M=\frac{9}{32} \sqrt{2}$ is indeed the smallest constant satisfying the inequality, with equality for any triple ( $a, b, c$ ) proportional to ( $1-\frac{3}{2} \sqrt{2}, 1,1+\frac{3}{2} \sqrt{2}$ ), up to permutation.