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Saturday, December 17, 2016

Given the positive real numbers a,b,c and x,y,z satisfying the condition: a+x=b+y=c+z=1 Prove the inequality (abc+xyz)(1ay+1bz+1cx)3.

Given the positive real numbers a,b,c and x,y,z satisfying the condition:

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=1a,y=1b,z=1c

(abc+xyz)(1ay+1bz+1cx)

=(abc+(1a)(1b)(1c))(1a(1b)+1b(1c)+1c(1a))

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

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

=(1a(1b)b(1c)c(1a))(1a(1b)+1b(1c)+1c(1a))

=1a(1b)+1b(1c)+1c(1a)1a(1b)(1b(1c)+1c(1a))

1b(1c)(1a(1b)+1c(1a))1c(1a)(1a(1b)+1b(1c))

\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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