I can't get it in that form!

Algebra Level 5

Let x , y x,y and z z be positive reals satisfying x + y + z + x y + y z + x z = 6051 2 . x+y+z+xy+yz+xz=\frac{6051}{2}. Find the minimum of the expression below. x 2 y z + y 2 z x + z 2 x y \dfrac{x^{2}y}{z}+\dfrac{y^{2}z}{x}+\dfrac{z^{2}x}{y}


The answer is 2931.7.

Son Nguyen by Son Nguyen , 20, Vietnam

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2 solutions

Jon Haussmann
Jan 22, 2016

Problem: Let x , y x,y and z z be positive reals satisfying x + y + z + x y + y z + x z = 6051 2 . x+y+z+xy+yz+xz=\frac{6051}{2}. Find the minimum of the expression below. x 2 y z + y 2 z x + z 2 x y \dfrac{x^{2}y}{z}+\dfrac{y^{2}z}{x}+\dfrac{z^{2}x}{y}

After some numerical testing, the smallest value I can get is approximately 2931.71, which occurs when x = y = z = 4035 − 1 2 x = y = z = \frac{\sqrt{4035} - 1}{2} . If someone can find a smaller value, please post.

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I concur with your result, Jon. I did it with AM-GM and then checked with WolframAlpha Pro and Mathematica - all three gave me exactly the same numbers as you found. I see, just now, that Calvin Lin has changed the answer to 2931.71 and those who answered 2931 or 2932 will get credit.

Bob Kadylo - 5 years, 2 months ago

How does a genius like you do numerical testing? Just curious.

Sal Gard - 5 years ago

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I just use a computer to test a lot of random values. Nothing very exciting.

Jon Haussmann - 4 years, 12 months ago

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Thanks for the response. Do you know any good resources or methods for synthetic geometry. I am having trouble finding resources and I would like to start solving Xuming Liang's problems (as you already have). Thanks.

Sal Gard - 4 years, 12 months ago

Am I the only one trying a = b = c a = b = c first when dealing with maximum / minimum problems ? Hehehehe

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