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Olimpiade Arsip | Catatan Nurkholis

Problem – $w$ dan $\dfrac{1}{w}$

Problem: Tentukan semua solusi real $x$, $y$, $z$, $w$ pada sistem: \begin{eqnarray*} x+y+z&=&w\\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}&=&\frac{1}{w} \end{eqnarray*} Solusi: Samakan penyebut pada persamaan ke dua diperoleh: $$\dfrac{yz+xz+xy}{xyz}=\dfrac{1}{w}$$ dengan mengalikan silang kemudian substitusi persamaan pertama pada $w$, didapat: $$(x+y+z)(yz+xz+xy)=xyz$$ yang ekuivalen dengan $$x^2y+x^2z+y^2x+y^2z+z^2x+z^2y+2xyz=0$$ dan dapat disederhanakan menjadi: $$(x+y)(x+z)(y+z)=0$$ dari sini dapat kita ambil solusinya (sebut saja $y+z=0$, $y=-z$) dan …

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