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

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

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