Olympiad selection grade 10th part II

Algebra Level 4

x 2 y 3 + y 2 z 3 + z 2 x 3 + ( x − 1 ) 2 + ( y − 1 ) 2 + ( z − 1 ) 2 \large x^2y^3+y^2z^3+z^2x^3+(x-1)^2+(y-1)^2+(z-1)^2

Given that x , y x,y and z z are positive reals satisfying x y + y z + x z = 3 xy+yz+xz=3 , find the minimum value of the expression above.

Undentifined 3 2 4 6 1 5 0

Son Nguyen by Son Nguyen , 20, Vietnam

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1 solution

Kushal Dey
Dec 7, 2020

Note that (4x²y³+y²z³+2z²x³)/7>= x²y²z by AM-GM inequality. Thus x²y³+y²z³+z²x³>=xyz(xy+yz+zx)=3xyz (by adding other 2 similar equations) . Now this is achieved when x=y=z, which in our case is 1 and (x-1)² and other two also attain minimum value at that point

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