Difference between revisions of "1985 OIM Problems/Problem 4"
(Created page with "== Problem == If <math>x \ne 1</math>, <math>y \ne 1</math>, <math>x \ne y</math>, and: <cmath>\frac{yz-x^2}{1-x}=\frac{xz-y^2}{1-y}</cmath> Prove that both fractions are equ...") |
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Prove that both fractions are equal to <math>x+y+z</math>. | Prove that both fractions are equal to <math>x+y+z</math>. | ||
| + | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
| + | |||
== Solution == | == Solution == | ||
| − | {{ | + | Because <math>x\ne y</math>, by ratio equalities, <math>\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</math>. |
| + | |||
| + | == See also == | ||
| + | https://www.oma.org.ar/enunciados/ibe1.htm | ||
Latest revision as of 22:50, 8 April 2024
Problem
If 
, 
, 
, and:
![\[\frac{yz-x^2}{1-x}=\frac{xz-y^2}{1-y}\]](http://latex.artofproblemsolving.com/d/1/7/d17d18b9b486581e704fdcc8624cdc663b7e8c64.png)
Prove that both fractions are equal to 
.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
Because 
, by ratio equalities, 
.
