Let a>b>c>1. How to find solutions in positive numbers of the following system? {ax>y+z, by>x+z, cz>x+y

Spactapsula2l

Spactapsula2l

Answered question

2022-09-09

Let a > b > c > 1. How to find solutions in positive numbers of the following system?
{ a x > y + z b y > x + z c z > x + y

Answer & Explanation

Mateo Tate

Mateo Tate

Beginner2022-09-10Added 18 answers

a , b , c are given and you want to find x , y , z > 0 satisfying your system. In terms of x, we have
min ( b y z , c z y ) > x > y / a + z / a
In order for this interval for x to be nonempty, we need
( b 1 / a ) y > ( 1 + 1 / a ) z ( c 1 / a ) z > ( 1 + 1 / a ) y
So (noting that c > 1 > 1 / a)
a b 1 a + 1 y > z > a + 1 a c 1 y
Finally, to make the interval for z be nonempty, we need
a b 1 a + 1 > a + 1 a c 1
or equivalently
a b c > a + b + c + 2
and then any y > 0 will do, with z and then x in the intervals given above.
Nadia Smith

Nadia Smith

Beginner2022-09-11Added 3 answers

{ a x > y + z b y > x + z c z > x + y
So,
a x + b y + c z > 2 ( x + y + z )
If it is an identity in x, y and z then this always holds for
a > b > c > 2 ; x , y , z > 0 x , y , z R

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