Cho x, y, z thỏa mãn (1/x+1/y+1/z)/(1/x+y+z)=1. tính giá trị biểu thức B=(x^21+y^21)(y^11+z^11)(z^2017+x^2017)

Đáp án:

Ta có : 

`(1/x + 1/y + 1/z)/[(1/(x + y + z)) = 1`

`=> 1/x + 1/y + 1/z = 1/(x + y + z)`

`=> 1/x + 1/y + 1/z –  1/(x + y + z) = 0`

`=>(  1/x + 1/y) + [1/z –  1/(x + y + z)] = 0`

`=> y/(xy) + x/(xy) + (x + y + z)/(z(x + y + z)) – z/(z(x + y + z)) = 0`

`=> (x + y)/(xy) + (x + y)/(z(x + y + z)) = 0`

`=> [(x + y)z(x + y + z) + (x + y)xy]/[xyz(x + y + z)] = 0`

`=> (x + y)z(x + y + z) + (x + y)xy = 0`

`=> (x + y)[z(x + y + z) + xy] = 0`

`=> (x + y)(zx + zy + z^2 + xy) = 0`

`=> (x + y)[(zx + z^2) + (zy + xy)]= 0`

`=> (x + y)[z(x + z) + y(z + x)] = 0`

`=> (x + y)(x + z)(y + z) = 0`

th1 : `x + y = 0 =>y = -x`

`=> B =(x^{21}+y^{21})(y^{11}+z^{11})(z^{2017}+x^{2017})`

`= [x^{21} – x^{21}].(y^{11}+z^{11})(z^{2017}+x^{2017})`

`= 0. (y^{11}+z^{11})(z^{2017}+x^{2017})`

`= 0`

th2 : `x + z = 0 => x = -z`

`=> B =(x^{21}+y^{21})(y^{11}+z^{11})(z^{2017}+x^{2017})`

` = (x^{21}+y^{21})(y^{11}+z^{11})(z^{2017} – z^{2017})`

`= (x^{21}+y^{21})(y^{11}+z^{11}) . 0`

`= 0`

th3 : `y + z = 0 => z = -y`

làm tương tự `=> B = 0`

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