Problem

Source: Serbia JBMO TST 2024 P2

Tags: inequalities



Let $a, b, c$ be positive reals such that $ab+bc+ca=\frac{3}{4}$. Show that $$(a+b+c)^6 \geq (\frac{9} {8})^3(1+(a+b)^2)(1+(b+c)^2)(1+(c+a)^2).$$When does equality hold?