Lời giải.
Ta có: `2021+y^2=xy+yz+zx+y^2=(y+x)(y+z)`
Tương tự: `2021+x^2=(x+y)(x+z)`
`2021+z^2=(z+x)(z+y).`
Suy ra:
`\sqrt{{(2021+y^2)(2021+z^2)}/{2021+x^2}}=\sqrt{{(y+x)(y+z)(z+y)(z+x)}/{(x+y)(x+z)}}=\sqrt{(y+z)^2}=y+z` (vì `y>0,z>0`)
`=>x\sqrt{{(2021+y^2)(2021+z^2)}/{2021+x^2}}=x(y+z)`
Tương tự: `y\sqrt{{(2021+z^2)(2021+x^2)}/{2021+y^2}}=y(x+z)`
`z\sqrt{{(2021+x^2)(2021+y^2)}/{2021+z^2}}=z(x+y)`
`=>P=x(y+z)+y(x+z)+z(x+y)=xy+xz+yx+yz+zx+zy=2(xy+yz+zx)=2.2021=4042.`