Problem 40

Algebra Level 5

8 x y z − 3 x 3 y 2 + z 2 \large 8xyz-\dfrac{3x^3}{y^2+z^2}

Given that x , y x,y and z z are positive reals satisfying x 2 + y 2 + z 2 = x y + 10 y z + x z x^2+y^2+z^2=xy+10yz+xz , find the minimum value of the expression above.

Set .


The answer is -64.

View wiki
Son Nguyen by Son Nguyen , 20, Vietnam

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Anand Raj
May 28, 2019

Since the second equation is symmetric in y and z, I assumed y = z and after solving you will get x = 4y = 4z. Substituting in the first equation we will get a simple cubic whose minima (and minimum as x, y, and z are positive) occurs at (4, 1, 1). Thus the answer is -64.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...