a+x=b+y=c+z=1
Prove the inequality (abc+xyz)(1ay+1bz+1cx)≥3.
My solution:
Rewrite the intended LHS of the inequality strictly in terms of a, b and c, we have:
x=1−a,y=1−b,z=1−c
(abc+xyz)(1ay+1bz+1cx)
=(abc+(1−a)(1−b)(1−c))(1a(1−b)+1b(1−c)+1c(1−a))
=(abc+1+ab+bc+ca−(a+b+c)−abc)(1a(1−b)+1b(1−c)+1c(1−a))
=(1+ab+bc+ca−(a+b+c))(1a(1−b)+1b(1−c)+1c(1−a))
=(1−a(1−b)−b(1−c)−c(1−a))(1a(1−b)+1b(1−c)+1c(1−a))
=1a(1−b)+1b(1−c)+1c(1−a)−1−a(1−b)(1b(1−c)+1c(1−a))
−1−b(1−c)(1a(1−b)+1c(1−a))−1−c(1−a)(1a(1−b)+1b(1−c))
\displaystyle =\frac{1-b(1-c)-c(1-a)}{a(1-b)}+\frac{1-a(1-b)-c(1-a)}{b(1-c)}+\frac{1-a(1-b)-b(1-c)}{c(1-a)}-3
\displaystyle =\frac{1-b+bc-c+ca}{a(1-b)}+\frac{1-a+ab-c+ac}{b(1-c)}+\frac{1-a+ab-b+bc}{c(1-a)}-3
\displaystyle =\frac{1-b-c(1-b)+ca}{a(1-b)}+\frac{1-c-a(1-c)+ab}{b(1-c)}+\frac{1-a-b(1-a)+bc}{c(1-a)}-3
\displaystyle =\frac{(1-b)(1-c)+ca}{a(1-b)}+\frac{(1-a)(1-c)+ab}{b(1-c)}+\frac{(1-b)(1-a)+bc}{c(1-a)}-3
\displaystyle =\frac{1-c}{a}+\frac{c}{1-b}+\frac{1-a}{b}+\frac{a}{1-c}+\frac{1-b}{c}+\frac{b}{1-a}-3
\displaystyle \ge 6\sqrt[6]{\frac{1-c}{a}\cdot \frac{c}{1-b}\cdot\frac{1-a}{b}\cdot\frac{a}{1-c}\cdot\frac{1-b}{c}\cdot\frac{b}{1-a}}-3 (By the AM-GM inequality, with 1-a,\,1-b,\,1-c are all positive)
\displaystyle = 6-3
\displaystyle = 3 (Q.E.D.)
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