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and 
be two isogonal conjugate with respect to triangle 
. Let 
and 
be two points lying on the circle 
such that 
and 
are perpendicular and parallel to bisector of 
, respectively. Prove that 
and 
bisect two arcs 
containing 
and not containing , respectively, of 
.
be a triangle such that 
and 
. The tangent at to the circumcircle 
of triangle meets the line at 
. Let 
be the reflection of in the line , let 
be the foot of the perpendicular from to 
, and let 
be the midpoint of the segment 
. Let the line 
intersect the circle again at 
.
is tangent to the circumcircle of triangle 
.![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](http://latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers 
, 
, 
, each different from 
, and satisfying 
., , , each different from , and satisfying .
there exists an -digit number divisible by 
all of whose digits are odd.
with 
is integer
can be prime only if 
so 
has at most two elements has integer root and non integer root. Then 
where 
is second non-integer root is prime then or 
is prime, then can be 
or 
then both 
and can not be integer at same time, because their difference is less than so has at most three elements
has integer root 
and 
are prime values, so can have 
values
