BdMO National Secondary 2010#3

[samiul_samin]

Solve for real $x$:$\dfrac {|x^2-1|}{x-2}=x$

[samiul_samin]

$\dfrac {|x^2-1|}{x-2}=x \Rightarrow |x^2-1|=x^2-2x $

$x^2-2x=x^2+1\Rightarrow 2x=1\Rightarrow x=\dfrac 12$
Or,
$x^2-2x=-x^2+1\Rightarrow 2x^2-2x-1=0\Rightarrow x=\dfrac {1+\sqrt 3}{2},\dfrac {1-\sqrt 3}{2}$
From these $3$ values only $\dfrac {1-\sqrt 3}{2}$ is the answer for real $x$.

[samiul_samin]

Solve for real $x$:$\dfrac {|x^2-1|}{x-2}=x$

This is German National Olympiad 2006 2nd round Problem 1.