Tìm 3 số x, y, z thỏa mãn điều kiện: 2018x – y2 = 2018y – z2 = 2018z – x2 = 2017.

Tìm 3 số x, y, z thỏa mãn điều kiện:

2018x – y² = 2018y – z² = 2018z – x² = 2017.

Ta có: $\left\{\begin{array}{l}2018x-y^2=2018y-z^2\\ 2018y-z^2=2018z-x^2\\ 2018z-x^2=2018x-y^2\end{array}\right.\iff \left\{\begin{array}{l}2018\left(x-y\right)=\left(y-z\right)\left(y+z\right)\left(1\right)\\ 2018\left(y-z\right)=\left(z-x\right)\left(z+x\right)\left(2\right)\\ 2018\left(z-x\right)=\left(x-y\right)\left(x+y\right)\left(3\right)\end{array}\right.\left\{\begin{array}{l}2018x-y^2=2018y-z^2\\ 2018y-z^2=2018z-x^2\\ 2018z-x^2=2018x-y^2\end{array}\right.\iff \left\{\begin{array}{l}2018\left(x-y\right)=\left(y-z\right)\left(y+z\right)\left(1\right)\\ 2018\left(y-z\right)=\left(z-x\right)\left(z+x\right)\left(2\right)\\ 2018\left(z-x\right)=\left(x-y\right)\left(x+y\right)\left(3\right)\end{array}\right.$

Lấy (1) . (2) . (3) ta được:

2018³ . (x – y)(y – z)(z – x) = (x – y)(y – z)(z – x)(x + y)(y + z)(z + x)

⇔ 2018³ = (x + y)(y + z)(z + x)

⇔ x + y = y + z = z + x = 2018

⇔ x = y = z

Theo đề bài, ta có:

2018x – y² = 2018y – z² = 2018z – x² = 2017

⇔ 2018x – x² = 2018y – y² = 2018z – z² = 2017

Có: 2018x – x² = 2017

⇔ x² – 2018x + 2017 = 0

⇔ x² – x – 2017x + 2017 = 0

⇔ x(x – 1) – 2017(x – 1) = 0

⇔ (x – 2017)(x – 1) = 0

⇔ x = 2017 hoặc x = 1

Vậy x = y = z = 2017 hoặc x = y = z = 1.