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If a, b, c are positive then prove that ((1 + a)(1 + b)(1 + c))7 > 77 a4b4c4.

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(1 + a)(1 + b)(1 + c) = 1 + a + b + c + ab + bc + ca + abc

> a + b + c + ab + bc + ca + abc

≥  7(a . b . c . ab . bc . ca . abc)1/7 (∴ A ≥  G)

i.e. (1 + a)(1 + b)(1 + c) ≥ 1 + 7(a4b4c4)1/7 > 7(a4b4c4)1/7

((1 + a)(1 + b)(1 + c))7 > 77 a4b4c4

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