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be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point 
is chosen on 
such taht 
and point 
is the foot of the perpendicular from to 
. Ray 
mets again at 
. Point 
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
. Prove that lies on .
, so that the equation 
has two real roots, so that the one is the square of the other.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both and 
. Integers and are coprime if 
.
such that: ![\[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]](http://latex.artofproblemsolving.com/e/c/f/ecf7fb3c2805290f2cb07b7fdd7c73e1c7a55b99.png)
on the plane. For every 
segment for which 
lies on the ray 
, and 
lies on the ray 
, consider a point 
on the segment such that 
. Find the geometric locus of such points .
on the plane. For every segment for which lies on the ray , and lies on the ray , consider a point on the segment such that . Find the geometric locus of such points .
