Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.9 AIME Find the answer

The unknown real numbers $x, y, z$ satisfy the equations $\frac{x+y}{1+z}=\frac{1-z+z^{2}}{x^{2}-x y+y^{2}} ; \quad \frac{x-y}{3-z}=\frac{9+3 z+z^{2}}{x^{2}+x y+y^{2}}$ Find $x$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

$\sqrt[3]{14}$ Cross-multiplying in both equations, we get, respectively, $x^{3}+y^{3}=$ $1+z^{3}, x^{3}-y^{3}=27-z^{3}$ . Now adding gives $2 x^{3}=28$ , or $x=\sqrt[3]{14}$ .

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