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Let x, y, z be real numbers such that . Prove that -------------------------------------------------------------------------------------------------------------

  1. Let x, y, z be real numbers such that x^2 + y^2 + z^2 - 2xyz = 1 . Prove that (1+x)(1+y)(1+z) \le 4 + 4xyz
  2. -------------------------------------------------------------------------------------------------------------

Grade:11

1 Answers

mycroft holmes
272 Points
8 years ago
Using the standard substitution, x = cos A, y = cos B, z = cos C, where A,B,C are angles of a triangle, we have to prove that
 
(1+cos A)(1+ cos B)(1+cos C)
 
We have (1+cos A)(1+cos B) =0 which is true)
 
Similarly (1+cos A cos B)(1+cos C)
 
Thus we have (1+cos A)(1+ cos B)(1+cos C)
 
 

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