[/quote]
,
and 
Prove that 
be real numbers such that 
.
Prove that 
and 
Prove that
Solution:
and 
then 
and 
.
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
Prove that
Prove that
and 
intersect at points 
and 
.
and 
,
intersects 
,
intersects 
.
is the circumcenter of 
,
.
and 
Prove that
be a triangle with incenter 
,
and 
are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side 
and 
are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
,
past 
past 
and let 
and 
be permutations of 
.
such that 
for all 
in 
is the number of possible pairs of permutations 
,
be pairwise coprime positive integers with 
being prime and 
.![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
,
units horizontally (left or right); or
and 
,
to 
using at most 
moves.
Y by
On the bisectors of angles 
of the convex quadrilateral 
taken points 
, respectively, so that line 
is parallel to 
, line 
is parallel to 
and line 
is parallel to 
. Prove that
a) the line
is parallel to 
;
b)
, if additionally it is known that 
.
of the convex quadrilateral 
taken points 
, respectively, so that line 
is parallel to 
, line 
is parallel to 
and line 
is parallel to 
. Prove thata) the line
is parallel to 
;b)
, if additionally it is known that 
.
