If x,y,z are positive numbers, then prove that (x)/(x+y) + (y)/(y+z) +(z)/(z+x) <= 2 Though I have solved a lot of problems on AM-GM inequality, I am unable to solve this one. I am also not showing my working because I do not think that they will be of any help.

amhailim

amhailim

Answered question

2022-09-21

If x , y , z are positive numbers, then prove that x x + y + y y + z + z z + x 2
Though I have solved a lot of problems on AM-GM inequality, I am unable to solve this one. I am also not showing my working because I do not think that they will be of any help.

Answer & Explanation

vidovitogv5

vidovitogv5

Beginner2022-09-22Added 10 answers

c y c x x + y c y c x + z x + y + z = 2
Alexus Deleon

Alexus Deleon

Beginner2022-09-23Added 1 answers

Solution by C-S.
We need to prove that:
c y c ( x x + y 1 ) 1
or
c y c y x + y 1 ,
which is true because by C-S
c y c y x + y = c y c y 2 x y + y 2 ( x + y + z ) 2 c y c ( x y + y 2 ) 1.
But the previous idea is still better:
c y c y x + y c y c y x + y + z = 1.

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