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1985 OIM Problems/Problem 4

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Problem

If $x \ne 1$, $y \ne 1$, $x \ne y$, and: \[\frac{yz-x^2}{1-x}=\frac{xz-y^2}{1-y}\] Prove that both fractions are equal to $x+y+z$.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

Because $x\ne y$, by ratio equalities, $\frac{yz-x^2}{1-x}=\frac{xz-y^2}{1-y}=\frac{yz-x^2-xz+y^2}{-x+y}=\frac{z(x-y)+(x+y)(x-y)}{x-y}=x+y+z$.

See also

https://www.oma.org.ar/enunciados/ibe1.htm

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