Problem 3

Determine the least real number

M

such that the inequality

\left| ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})\right|\leq M(a^{2}+b^{2}+c^{2})^{2}

holds for all real numbers

a

b

and

c

Step 1 of 8: Factor the cyclic sum as a product of linear factors

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In plain words

Treating the messy cyclic sum as a polynomial in just one of its variables turns an algebraic identity-hunt into simple root-checking.

ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)=(b-c)(a-b)(a-c)(a+b+c)

Detailed analysis

View the left side as a cubic in $a$ with $b$ and $c$ fixed; its coefficient of $a^3$ is $b-c$ . Since $a=b$ , $a=c$ and $a=-b-c$ each make it vanish, it must equal $(b-c)(a-b)(a-c)(a+b+c)$ .