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If a , b , c , are positive real numbers...

If `a , b , c ,` are positive real numbers, then prove that (2004, 4M) `{(1+a)(1+b)(1+c)}^7>7^7a^4b^4c^4`

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Here, `(1 + a) (1 + b) (1 +c)`
`= 1 + a + b + c + ab + bc + ca + abc`....(i)
since, `(a + b+ c ab + bc + ca + abc)/(7) ge (a^(4) b^(4) c^(4))^(1//7)`[using `AM ge GM`]
`rArr a + b + c + ab + bc + ca + abc ge 7 (a^(4) b^(4) c^(4))^(1//7)`
`rArr 1 + a + b + c + ca + abc gt 7 (a^(4) b^(4) c^(4))^(1//7)`...(ii) ltbr. From Eqs. (i) and (ii), we get
`(1 + a) (1 + b) (1 +c) gt 7(a^(4) b^(4) c^(4))^(1//7)`
or `{(1 +a) (1 +b) (1 +c)}^(7) gt 7^(7) (a^(4) b^(4) c^(4))`
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