Saudi Arabia Mathematical Competitions

The inequality is equivalent to $$8(a+b+c)(ab+bc+ca)\le 9(a+b)(b+c)(c+a)$$ Using the identity $$(a+b+c)(ab+bc+ca)=(a+b)(b+c)(c+a)+abc$$ we get $$8(a+b)(b+c)(c+a)+8abc\le 9(a+b)(b+c)(c+a),$$ hence $8abc\le (a+b)(b+c)(c+a)$. Applying AM-GM inequality, it follows $$2\sqrt{ab}\le a+b,2\sqrt{bc}\le b+c,2\sqrt{ca}\le c+a,$$ and by multiplication $8abc\le (a+b)(b+c)(c+a)$. The equality holds if and only if $a=b=c$.