MathDB

1

Part of 2008 Mathcenter Contest

Problems(2)

x^2+xy+y^2=57, y^2+yz+z^2=84, z^2+zx+x^2=111

Source: Mathcenter Contest / Oly - Thai Forum 2008 R1 p1 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/10/2022
Given x,y,z∈R+x,y,z\in \mathbb{R} ^+ , that are the solutions to the system of equations : x2+xy+y2=57x^2+xy+y^2=57 y2+yz+z2=84y^2+yz+z^2=84 z2+zx+x2=111z^2+zx+x^2=111 What is the value of xy+3yz+5zxxy+3yz+5zx?
(maphybich)
algebrasystem of equationsSystem
sumx/\sqrt {x + y} >= \sqrt [4]{\frac {27(yz + zx + xy)}{4}}

Source: Mathcenter Contest / Oly - Thai Forum 2008 R2 p1 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/11/2022
Let x,y,zx,y,z be a positive real numbers. Prove that xx+y+yy+z+zz+x≥27(yz+zx+xy)44\frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac { z}{\sqrt {z + x}}\geq\sqrt [4]{\frac {27(yz + zx + xy)}{4}}
(dektep)
algebrainequalities