1 Rabbit-2 cage

Algebra Level 5

x y + y z + x z − 2 x y z xy + yz + xz - 2xyz

If x , y x,y and z z are positive real numbers such that x 2 + y 2 + z 2 + 2 x y z = 1 x^2 + y^2 + z^2 + 2xyz = 1 , find the maximum value of the expression above.


The answer is 0.5.

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Son Nguyen by Son Nguyen , 20, Vietnam

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1 solution

Son Nguyen
Oct 25, 2015

If in 3 number (2x-1),(2y-1),(2z-1) are always exists two possitive Real number. I f ( 2 x − 1 ) ( 2 y − 1 ) ≥ 0 ⇒ 2 ( x + y ) − 4 x y ≤ 1 ⇒ z ( x + y ) − 2 x y z ≤ z 2 If (2x-1)(2y-1)\geq 0\Rightarrow 2(x+y)-4xy\leq 1\Rightarrow z(x+y)-2xyz\leq \frac{z}{2} A n d : 1 − z 2 = 2 x y z + x 2 + y 2 ≥ 2 x y + 2 x y z = 2 x y ( z + 1 ) ⇒ x y ≤ 1 − z 2 And:1-z^2=2xyz+x^2+y^2\geq 2xy+2xyz=2xy(z+1)\Rightarrow xy\leq \frac{1-z}{2} → S ≤ 1 − z 2 + z 2 = 1 2 \rightarrow S\leq \frac{1-z}{2}+\frac{z}{2}=\frac{1}{2} ====>0.5(THE ANSWER) Actually it's dirichlet in inequality

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nice question and answer!

Pi Han Goh - 5 years, 7 months ago

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We can also use Cauchy-Schwarz Inequality and then AM-GM inequality to get the answer.

Kushagra Sahni - 5 years, 7 months ago

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Can you post a solution as well? I'm not getting the answer~

Pi Han Goh - 5 years, 7 months ago

Actually Is dirichlet in Inequality haha

Son Nguyen - 5 years, 7 months ago

Post a solution with AM GM PLEASE

Aakash Khandelwal - 5 years, 7 months ago

I used Directle in Inequality That I called"One Rabbit-Two cage"

Son Nguyen - 5 years, 7 months ago

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