Solve Inequality Challenge: Prove $35\sqrt{55}+55\sqrt{77}+77\sqrt{35}\gt 2310$.

[anemone]

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

 

[kaliprasad]

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

Spoiler

using AM GM ineqality we have unless a b and c same

$ab\sqrt{bc} + bc\sqrt{ca} + ca\sqrt{ab } \gt\ 3abc$
putting $a= 7, b= 5,c =11$ we get
$35 \sqrt{55} + 55\sqrt{77} + 77\sqrt{35 } \gt\ 3 * 7 * 5 * 11\cdots(1)$
putting $a= 5, b= 7,c =11$ we get
$35 \sqrt{77} + 77\sqrt{55} + 55\sqrt{35 } \gt\ 3 * 7 * 5 * 11\cdots(2)$
adding above we get
$35 \sqrt{55} + 55\sqrt{77} + 77\sqrt{35 } + 35 \sqrt{77} + 77\sqrt{55} + 55\sqrt{35 } \gt 3 * 5 * 7 * 11 * 2$ or 2310

 

[Albert1]

Prove $35\sqrt{55}+55\sqrt{77}+77\sqrt{35}+35\sqrt{77}+55\sqrt{35}+77\sqrt{55}\gt 2310---(1)$.

I also use AM-GM inequality:

Spoiler

$(1)>6\times\sqrt[6]{35^3\times 55^3 \times 77^3}=6\sqrt {35\times 55 \times 77}=2310$