An algebra problem by Priyanshu Mishra

Algebra Level 5

If $(x,y,z)$ are non-negative reals and $xy + yz + zx = 1$ ,

The maximum value of the expression below is of the form $\large \frac { a }{ b\sqrt { c } }$ .

$\large\ x\left( 1-{ y }^{ 2 } \right) \left( 1-{ z }^{ 2 } \right) + y\left( 1-{ z }^{ 2 } \right) \left( 1-{ x }^{ 2 } \right) + z\left( 1-{ x }^{ 2 } \right) \left( 1-{ y }^{ 2 } \right)$

$a,b,c$ are all positive integers with $a,b$ coprime and $c$ square-free.

Find $a + b + c$ .

The answer is 10.

1 solution

Rajen Kapur
Feb 3, 2017

Given expression: $\sum_{cyl} x(1-(y^2+z^2)+y^2z^2)=\sum x-\sum_{cyl}(xy^2)+xyz(xy+yz+zx)=4xyz$ Using $\sum x=(x+y+z)(xy+yz+zx)=\sum (xy^2)+3xyz$ Now A.M. - G.M.: $\frac {xy+yz+zx}{3}\geq(xyz)^\frac {3}{2}$ We get $a=4, b=3, c=3$ , hence answer 10.

An algebra problem by Priyanshu Mishra

The answer is 10.

1 solution

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