a-b-c-gt-0-amp-abc-1-prove-that-1-1-a-b-1-1-b-c-1-1-c-a-1-

Question Number 195820 by York12 last updated on 11/Aug/23

a, b, c > 0 & a b c = 1, p r o v e t h a t

\frac{1}{1 + a + b} + \frac{1}{1 + b + c} + \frac{1}{1 + c + a} ⩽ 1

Answered by CrispyXYZ last updated on 12/Aug/23

let a = x^{3}, b = y^{3}, c = z^{3}, then x y z = 1 .

x^{3} + y^{3} - x^{2} y - y^{2} x = {(x - y)}^{2} (x + y) ⩾ 0

\Rightarrow x^{3} + y^{3} ⩾ x^{2} y + y^{2} x

\frac{1}{1 + a + b} = \frac{1}{1 + x^{3} + y^{3}}

⩽ \frac{1}{1 + x^{2} y + y^{2} x}

= \frac{1}{x y z + x^{2} y + y^{2} x} = \frac{1}{x y (x + y + z)} = \frac{z}{x + y + z}

Similarly,

\frac{1}{1 + b + c} ⩽ \frac{x}{x + y + z}, \frac{1}{1 + c + a} ⩽ \frac{y}{x + y + z} .

In conclusion,

\frac{1}{1 + a + b} + \frac{1}{1 + b + c} + \frac{1}{1 + a + c} ⩽ \frac{x + y + z}{x + y + z} = 1

Commented by York12 last updated on 13/Aug/23

t b a n k s

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