Question

Đã trả lời bởi chuyên gia

Answer 1

Ta có: $\{\begin{cases} 2018 x - y^{2} = 2018 y - z^{2} \\ 2018 y - z^{2} = 2018 z - x^{2} \\ 2018 z - x^{2} = 2018 x - y^{2} \end{cases} \Leftrightarrow \{\begin{cases} 2018 (x - y) = (y - z) (y + z) (1) \\ 2018 (y - z) = (z - x) (z + x) (2) \\ 2018 (z - x) = (x - y) (x + y) (3) \end{cases} \{\begin{cases} 2018 x - y^{2} = 2018 y - z^{2} \\ 2018 y - z^{2} = 2018 z - x^{2} \\ 2018 z - x^{2} = 2018 x - y^{2} \end{cases} \Leftrightarrow \{\begin{cases} 2018 (x - y) = (y - z) (y + z) (1) \\ 2018 (y - z) = (z - x) (z + x) (2) \\ 2018 (z - x) = (x - y) (x + y) (3) \end{cases}$

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.

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Answer 2

Ta có: $\{\begin{cases} 2018 x - y^{2} = 2018 y - z^{2} \\ 2018 y - z^{2} = 2018 z - x^{2} \\ 2018 z - x^{2} = 2018 x - y^{2} \end{cases} \Leftrightarrow \{\begin{cases} 2018 (x - y) = (y - z) (y + z) (1) \\ 2018 (y - z) = (z - x) (z + x) (2) \\ 2018 (z - x) = (x - y) (x + y) (3) \end{cases} \{\begin{cases} 2018 x - y^{2} = 2018 y - z^{2} \\ 2018 y - z^{2} = 2018 z - x^{2} \\ 2018 z - x^{2} = 2018 x - y^{2} \end{cases} \Leftrightarrow \{\begin{cases} 2018 (x - y) = (y - z) (y + z) (1) \\ 2018 (y - z) = (z - x) (z + x) (2) \\ 2018 (z - x) = (x - y) (x + y) (3) \end{cases}$