Cho ` \frac{x^2}{x + y} + \frac{y^{2}}{y + z} + \frac{z^2}{z + x} = 2017 ` Tính: ` \frac{y^2}{x + y} + \frac{z^2}{y + z} + \frac{x^2}{z + x} – 3 `

Đáp án:

 c1 .

Ta có

`x^2/(x + y) + y^2/(y + z) + z^2/(z + x) = 2017`

`<=> (x^2 – y^2 + y^2)/(x + y) + (y^2 – z^2 + z^2)/(y + z) + (z^2 – x^2 + x^2)/(z + x) = 2017`

`<=> x – y + y^2/(x + y) + y – z + z^2/(y + z) + z – x + x^2/(z + x) = 2017`

`<=> y^2/(x + y) + z^2/(z + x) + x^2/(z + x) = 2017`

`-> y^2/(x + y) + z^2/(z + x) + x^2/(z + x) – 3 = 2017 – 3 = 2014`

c2

Ta có

`x^2/(x + y) + y^2/(y + z) + z^2/(z + x) = 2017`

`<=> (x – x^2/(x + y)) + (y – y^2/(y + z)) + (z – z^2/(z + x)) = x + y + z – 2017`

`<=> (xy)/(x + y) + (yz)/(y + z) + (zx)/(z + x) = x + y + z – 2017`

`+) 2(x + y + z) = (x + y) + (y + z) + (z + x) = (x + y)^2/(x + y) + (y + z)^2/(y + z) + (z + x)^2/(z + x)`

`= x^2/(x + y) + (2xy)/(x + y) + y^2/(x + y) + y^2/(y + z) + (2yz)/(y + z) + z^2/(y + z) + z^2/(z + x) + (2zx)/(z + x) + x^2/(z + x)`

`= (y^2/(x + y) + z^2/(z + x) + x^2/(z + x)) + (x^2/(x + y) + y^2/(y + z) + z^2/(z + x)) + 2.[ (xy)/(x + y) + (yz)/(y + z) + (zx)/(z + x) ]`

`<=> y^2/(x + y) + z^2/(z + x) + x^2/(z + x) – 3 = 2(x + y + z) – 2.[ (xy)/(x + y) + (yz)/(y + z) + (zx)/(z + x) ] – (x^2/(x + y) + y^2/(y + z) + z^2/(z + x)) – 3`

`= 2(x + y + z) – 2.[(x + y + z) – 2017] – 2017 – 3`

`= 2(x + y + z) – 2(x + y + z) + 2.2017 – 2017 – 3`

`= 2.2017 – 2017 – 3`

`= 2017 – 3`

`= 2014`

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