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Algebra Level 4

Let x x , y y , z z > 0 and if x 2 x + y + z + y x + 2 y + z + z x + y + 2 z ≤ k l \dfrac { x }{ 2x + y + z } + \dfrac { y }{ x + 2y + z } + \dfrac { z }{ x + y + 2z } \le \dfrac { k }{ l } for coprime positive integers k k and l l . Find the value of k + l k + l .


The answer is 7.

Priyanshu Mishra by Priyanshu Mishra , 20, India

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2 solutions

James Wilson
Jan 20, 2018

This solution lacks proper justification. Perhaps it can be justified by an argument involving curvature alongside the AM-GM inequality. I set x 2 x + y + z = y x + 2 y + z = z x + y + 2 z \frac{x}{2x+y+z}=\frac{y}{x+2y+z}=\frac{z}{x+y+2z} . These equations can be manipulated to the pair ( x + y + z ) ( x − y ) = 0 (x+y+z)(x-y)=0 and ( x + y + z ) ( y − z ) = 0 (x+y+z)(y-z)=0 . Dividing by x + y + z x+y+z (since it is positive), gives x = y = z x=y=z . Then simply replacing y y and z z by x x in the original expression leads to the maximum value 3 4 \frac{3}{4} .

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Deepansh Jindal
Jul 10, 2016

@Chew-Seong Cheong sir please upload a solution....

1 Helpful 0 Interesting 0 Brilliant 0 Confused

@Deepansh Jindal We can use Jensen's over here...........

Aaghaz Mahajan - 2 years, 11 months ago

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