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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))

=1b(1c)c(1a)a(1b)+1a(1b)c(1a)b(1c)+1a(1b)b(1c)c(1a)3

=1b+bcc+caa(1b)+1a+abc+acb(1c)+1a+abb+bcc(1a)3

=1bc(1b)+caa(1b)+1ca(1c)+abb(1c)+1ab(1a)+bcc(1a)3

=(1b)(1c)+caa(1b)+(1a)(1c)+abb(1c)+(1b)(1a)+bcc(1a)3

=1ca+c1b+1ab+a1c+1bc+b1a3

661cac1b1aba1c1bcb1a3 (By the AM-GM inequality, with 1a,1b,1c are all positive)

=63

=3 (Q.E.D.)


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