Question

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Answer 1

If a,b,c are real and distinct, then show that
\[\huge a ^{2}(1+b ^{2})+b ^{2}(1+c ^{2})+c ^{2}(1+a ^{2}) > 6abc\]

Answer 2

\[\large a ^{2}(1+b ^{2})+b ^{2}(1+c ^{2})+c ^{2}(1+a ^{2}) > 6abc\]

Answer 3

@mathmate

Answer 4

@ganeshie8 help

Answer 5

which terms i should consider , i have tried many combinations none work :(

Answer 6

@ganeshie8

Answer 7

\[\large (a^2 + b^2 + c^2) + (a^2b^2 + b^2c^2 + c^2a^2) \]

Answer 8

when you open the brackets you get this

Answer 9

By AM-GM inequality :
\[\large \gt (3\sqrt[3]{a^2b^2c^2}) + (3\sqrt[3]{a^4b^4c^4}) \]

Answer 10

yes

Answer 11

\[\large \gt 3abc\left( (abc)^{\dfrac{-1}{3} } + (abc)^{\dfrac{1}{3}}\right) \]

Answer 12

still yes ? :)

Answer 13

got it

Answer 14

good :) the minimum value of x+1/x is 2

Answer 15

\[\large \gt 3abc\left( 2\right) \]

Answer 16

\[\large \gt 6abc \]

Answer 17

thank you

Answer 18

I am not quite sure how you got the second step

Answer 19

oh which step ?

Answer 20

By AM-GM inequality : after this

Answer 21

consider : a^2, b^2, c^2

AM = ?
GM = ?