Question

Если
$\displaystyle \frac{x+y}{z} + \frac{y+z}{x} + \frac{z+x}{y} = 2022$
То найдите значение

$\dfrac{(x+y)(y+z)(x+z)}{xyz} =?$

Answer 1

Відповідь:

2024

Покрокове пояснення:

$\displaystyle\frac{x+y}{z} +\frac{y+z}{x} +\frac{z+x}{y}=2022\\\displaystyle\frac{(x+y)xy}{zxy} +\frac{(y+z)yz}{xyz} +\frac{(z+x)zx}{yzx}=2022\\\\\frac{(x+y)xy+(y+z)yz+(z+x)zx}{xyz} =2022$

$\displaystyle\frac{(x+y)(y+z)(x+z)}{xyz} =\\=\frac{(xy+xz+y^2+yz)(x+z)}{xyz} =\\=\frac{x^2y+x^2z+xy^2+xyz+xyz+xz^2+y^2z+yz^2}{xyz}=\\=\frac{x^2y+xy^2+y^2z+yz^2+xz^2+x^2z+xyz+xyz}{xyz}=\\=\frac{(x+y)xy+(y+z)yz+(x+z)xz+2xyz}{xyz}=\\=\frac{(x+y)xy+(y+z)yz+(z+x)zx}{xyz}+\frac{2xyz}{xyz}=\\=2022+2=2024$