Đáp án:

Đặt 

`M = [(x(x + y))/(x - y) + (x(x + z))/(x - z)]/[1 + (y - z)^2/[(x - y)(x - z)]] + [(y(y + z))/(y - z) + (y(y + x))/(y - x)]/[1 + (z - x)^2/[(y - z)(y - x)]] + [(z(z + x))/(z - x) + (z(z + y))/(z - y)]/[1 + (x - y)^2/[(z - x)(z - y)]]`

 Ta có

`[(x(x + y))/(x - y) + (x(x + z))/(x - z)]/[1 + (y - z)^2/[(x - y)(x - z)]]`

`= [(x(x + y)(x - z))/((x - y)(x - z)) + (x(x + z)(x - y))/((x - y)(x - z))]/[[(x - y)(x - z)]/[(x - y)(x - z)] + (y - z)^2/[(x - y)(x - z)]]`

`= [(x(x + y)(x - z) + x(x + z)(x - y))/((x - y)(x - z))]/[[(x - y)(x - z) + (y - z)^2]/[(x - y)(x - z)]]`

`= [(x(x + y)(x - z) + x(x + z)(x - y))]/[(x - y)(x - z) + (y - z)^2]`

`= (x^3 + x^2y - x^2z - xyz + x^3 + x^2z - x^2y - xyz)/(x^2 - xy - xz + yz + y^2 - 2yz + z^2)`

`= (2x^3 - 2xyz)/(x^2 + y^2 + z^2 - xy - yz - zx)`

Tương tự 

`[(y(y + z))/(y - z) + (y(y + x))/(y - x)]/[1 + (z - x)^2/[(y - z)(y - x)]]`

`= (2y^3 - 2xyz)/(x^2 + y^2 + z^2 - xy - yz - zx)`

`[(z(z + x))/(z - x) + (z(z + y))/(z - y)]/[1 + (x - y)^2/[(z - x)(z - y)]]`

`= (2z^3 - 2xyz)/(x^2 + y^2 + z^2 - xy - yz - zx)`

Do đó

`M = [(x(x + y))/(x - y) + (x(x + z))/(x - z)]/[1 + (y - z)^2/[(x - y)(x - z)]] + [(y(y + z))/(y - z) + (y(y + x))/(y - x)]/[1 + (z - x)^2/[(y - z)(y - x)]] + [(z(z + x))/(z - x) + (z(z + y))/(z - y)]/[1 + (x - y)^2/[(z - x)(z - y)]]`

`= (2x^3 - 2xyz)/(x^2 + y^2 + z^2 - xy - yz - zx) + (2y^3 - 2xyz)/(x^2 + y^2 + z^2 - xy - yz - zx) + (2z^3 - 2xyz)/(x^2 + y^2 + z^2 - xy - yz - zx)`

`= (2x^3 - 2xyz + 2y^3 - 2xyz + 2z^3 - 2xyz)/(x^2 + y^2 + z^2 - xy - yz - zx)`

`= [2(x^3 + y^3 + z^3 - 3xyz)]/(x^2 + y^2 + z^2 - xy - yz - zx) (**)`

Ta có : 

`x^3 + y^3 + z^3 - 3xyz`

`= [(x + y)^3 + z^3] - [3xy(x + y) + 3xyz]`

`= (x + y + z)[(x + y)^2 - (x + y)z + z^2] - 3xy(x + y + z)`

`= (x + y + z)(x^2 + 2xy + y^2 - xz - yz + z^2 - 3xy)`

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

Thay `(*)` vào `(**)` ta được

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

`= 2(x + y + z)`

`-> [ [(x(x + y))/(x - y) + (x(x + z))/(x - z)]/[1 + (y - z)^2/[(x - y)(x - z)]] + [(y(y + z))/(y - z) + (y(y + x))/(y - x)]/[1 + (z - x)^2/[(y - z)(y - x)]] + [(z(z + x))/(z - x) + (z(z + y))/(z - y)]/[1 + (x - y)^2/[(z - x)(z - y)]] ] : (x + y + z)`

`= 2(x + y + z) : (x + y + z)`

`= 2`

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