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Algebra Difficulty 5.0 AIME Prove it Ibero-American Mathematical Olympiad

Problem:

The reals x,y,zx, y, z satisfy x≠1x \neq 1, y≠1y \neq 1, x≠yx \neq y, and yz−x21−x=xz−y21−y\dfrac{y z - x^{2}}{1 - x} = \dfrac{x z - y^{2}}{1 - y}. Show that yz−x21−x=x+y+z\dfrac{y z - x^{2}}{1 - x} = x + y + z.

Solution

Solution:

We have yz−x2−y2z+yx2=xz−y2−x2z+xy2y z - x^{2} - y^{2} z + y x^{2} = x z - y^{2} - x^{2} z + x y^{2}. Hence z(y−x−y2+x2)=−y2+xy2−x2y+x2z (y - x - y^{2} + x^{2}) = -y^{2} + x y^{2} - x^{2} y + x^{2}. Hence z=x+y−xyx+y−1z = \dfrac{x + y - x y}{x + y - 1}.

So yz=x+y+z−xy−xzy z = x + y + z - x y - x z, so yz−x2=x+y+z−x2−xy−xz=(x+y+z)(1−x)y z - x^{2} = x + y + z - x^{2} - x y - x z = (x + y + z)(1 - x), so yz−x21−x=x+y+z\dfrac{y z - x^{2}}{1 - x} = x + y + z.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.