I've been trying to understand where to take this math problem... I've tried substituting all of these terms below in the form of $BC$ since it is an equilateral triangle but that just gets messy and I cant figure out what else to try.
Triangle $ABC$ is equilateral, $L,M,N$ are points on $BC,CA$ and $AB$ respectively. Prove that $MA\cdot AN+NB\cdot BL+LC\cdot CM < BC^2$.
On
$$BC^3=(BL+LC)(CM+MA)(AN+NB)=$$ $$=BL\cdot CM\cdot AN+LC\cdot MA\cdot NB+LC\cdot CM\cdot AB+AN\cdot MA\cdot BC+NB\cdot BL\cdot AC>$$ $$>LC\cdot CM\cdot AB+AN\cdot MA\cdot BC+NB\cdot BL\cdot AC=BC(LC\cdot CM+AN\cdot MA+NB\cdot BL)$$ and we are done!
On
With words:
$$S(AMN)=\frac{AM\cdot AN\cdot \sin 60°}{2}=\frac{AM\cdot AN\cdot \sqrt{3}}{4}\\ S(CML)=\frac{CM\cdot CL\cdot \sin 60°}{2}=\frac{CM\cdot CL\cdot \sqrt{3}}{4}\\ S(BNL)=\frac{BN\cdot BL\cdot \sin 60°}{2}=\frac{BN\cdot BL\cdot \sqrt{3}}{4}\\ S(ABC)=\frac{AB\cdot BC\cdot \sin 60°}{2}=\frac{BC^2\cdot \sqrt{3}}{4}$$
But,
$$S(AMN)+S(CML)+S(BNL)<S(ABC)\to\\ AM\cdot AN+BN\cdot BL+CM\cdot CL<BC^2$$
Without words ${}{}{}{}{}{}{}$: