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An exercise from the Mathematics Olympiad with a solution , Prove that x²/y+y²/z+z²/x ≥ x+y+z

 

x and y and z are positive real numbers

Prove that :

\frac{x^{2}}{y}+\frac{y^{2}}{z}+\frac{z^{2}}{x}\geq x+y+z




solution:

we have (x-y)² ≥ 0
Means x²-2xy+y² ≥ 0 , Means x²+y² ≥ 2xy  ,
 Means  (x²+y²)/y ≥ 2xy/y , then  x²/y+y  ≥ 2x   (1)
In the same way we show that :  y²/z+z  ≥ 2y   (2)
                                                     z²/x+x ≥ 2z   (3)

We collect the inequalities 1 and 2 and 3 

 x²/y+y + y²/z+z + z²/x+x  2x+2y+2z

   so:

               x²/y + y²/z + z²/x  x+y+z


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