(Heuristic Solution) Prove $35\sqrt{55}+55\sqrt{77}+77\sqrt{35}+35\sqrt{77}+55\sqrt{35}+77\sqrt{55}\gt 2310$.

Prove $35\sqrt{55}+55\sqrt{77}+77\sqrt{35}+35\sqrt{77}+55\sqrt{35}+77\sqrt{55}\gt 2310$.

My solution:

Note that we could rewrite the given LHS of the inequality as follows:

$35\sqrt{55}+55\sqrt{77}+77\sqrt{35}+35\sqrt{77}+55\sqrt{35}+77\sqrt{55}$

$\small =5(7)\sqrt{5(11)}+5(11)\sqrt{7(11)}+7(11)\sqrt{5(7)}+5(7)\sqrt{7(11)}+5(11)\sqrt{5(7)}+7(11)\sqrt{5(11)}$

Now, we could further algebraically manipulating the expression by dividing (at the same time multiplying) them by $5(7)(11)$:

$35\sqrt{55}+55\sqrt{77}+77\sqrt{35}+35\sqrt{77}+55\sqrt{35}+77\sqrt{55}$

$\small=5(7)\sqrt{5(11)}+5(11)\sqrt{7(11)}+7(11)\sqrt{5(7)}+5(7)\sqrt{7(11)}+5(11)\sqrt{5(7)}+7(11)\sqrt{5(11)}$

$\scriptsize=(5)(7)(11)\left(\dfrac{5(7)\sqrt{5(11)}}{5(7)(11)}+\dfrac{5(11)\sqrt{7(11)}}{5(7)(11)}+\dfrac{7(11)\sqrt{5(7)}}{5(7)(11)}+\dfrac{5(7)\sqrt{7(11)}}{5(7)(11)}+\dfrac{5(11)\sqrt{5(7)}}{5(7)(11)}+\dfrac{7(11)\sqrt{5(11)}}{5(7)(11)}\right)$

$=(5)(7)(11)\left(\dfrac{\sqrt{5}}{\sqrt{11}}+\dfrac{\sqrt{11}}{\sqrt{7}}+\dfrac{\sqrt{7}}{\sqrt{5}}+\dfrac{\sqrt{7}}{\sqrt{11}}+\dfrac{\sqrt{5}}{\sqrt{7}}+\dfrac{\sqrt{11}}{\sqrt{5}}\right)$

$=(5)(7)(11)\left(\left(\dfrac{\sqrt{5}}{\sqrt{11}}+\dfrac{\sqrt{11}}{\sqrt{5}}\right)+\left(\dfrac{\sqrt{11}}{\sqrt{7}}+\dfrac{\sqrt{7}}{\sqrt{11}}\right)+\left(\dfrac{\sqrt{7}}{\sqrt{5}}+\dfrac{\sqrt{5}}{\sqrt{7}}\right)\right)$

Note that from the AM-GM inequality, we have:

$\dfrac{\sqrt{5}}{\sqrt{11}}+\dfrac{\sqrt{11}}{\sqrt{5}}=\dfrac{\sqrt{5}}{\sqrt{11}}+\dfrac{1}{\dfrac{\sqrt{5}}{\sqrt{11}}}\gt 2$

Similarly,

$\dfrac{\sqrt{11}}{\sqrt{7}}+\dfrac{\sqrt{7}}{\sqrt{11}}=\dfrac{\sqrt{11}}{\sqrt{7}}+\dfrac{1}{\dfrac{\sqrt{11}}{\sqrt{7}}}\gt 2$ and $\dfrac{\sqrt{7}}{\sqrt{5}}+\dfrac{\sqrt{5}}{\sqrt{7}}=\dfrac{\sqrt{7}}{\sqrt{5}}+\dfrac{1}{\dfrac{\sqrt{7}}{\sqrt{5}}}\gt 2$

Therefore, we obtain:

$35\sqrt{55}+55\sqrt{77}+77\sqrt{35}+35\sqrt{77}+55\sqrt{35}+77\sqrt{55}$

$=(5)(7)(11)\left(\left(\dfrac{\sqrt{5}}{\sqrt{11}}+\dfrac{\sqrt{11}}{\sqrt{5}}\right)+\left(\dfrac{\sqrt{11}}{\sqrt{7}}+\dfrac{\sqrt{7}}{\sqrt{11}}\right)+\left(\dfrac{\sqrt{7}}{\sqrt{5}}+\dfrac{\sqrt{5}}{\sqrt{7}}\right)\right)$

$\gt (5)(7)(11)(2+2+2)$

$\gt 2310$

And we're hence done.