attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
satisfying 
.
that satisfy 
with 
,
and area 
.
with diameter 
at 
.
,
and extend to meet 
such that 
and 
is the diameter of 
,
of 
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
.
lie on 
respectively, such that 
.
to meet the circumcircle of 
respectively. Let the circumcircles of 
meet at points 
.
to 
,
,![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
.
with 
,
.
to meet the circumcircle of 
at 
,
.
Y by chessgocube, Adventure10, Mango247
The trapezoid 
, of bases 
and 
, is inscribed in a circumference 
. Let 
a variable point of the arc of that does not contain 
or 
. We denote 
to the point of intersection of and 
, and let Z be the point of the segment 
such that 
. Prove that the measure of 
does not depend on the choice of 
, of bases 
and 
, is inscribed in a circumference 
. Let 
a variable point of the arc of that does not contain 
or 
. We denote 
to the point of intersection of and 
, and let Z be the point of the segment 
such that 
. Prove that the measure of 
does not depend on the choice of 
and radius 
.
with 
.![$\tan \angle AZX=\frac{(b-r)[b(d+r)+dr+r^{2}+vw]}{b[bv-dw-r(v+w)]}$](http://latex.artofproblemsolving.com/4/a/e/4ae0ef513276bf6195baa7759d47caba3f47302c.png)
.
