be positive real integer such that 
Prove that
such that 
and 
that verify the following equation for all 
:
is a prime number
be an acute scalene triangle with circumcenter 
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
be the circle with diameter 
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
Y by
In the quadrilateral 
is 
and the angles 
and 
are acute. The diagonals intersect in 
. The circumscribed circles of 
and 
intersect the line 
again at 
and 
respectively. Call 
the midpoint of ![$[ST]$](//latex.artofproblemsolving.com/e/5/d/e5d84c692bb0db11678ce289ae0ae6e1f6ba3d4a.png)
. Prove that 
is isosceles.
is 
and the angles 
and 
are acute. The diagonals intersect in 
. The circumscribed circles of 
and 
intersect the line 
again at 
and 
respectively. Call 
the midpoint of ![$[ST]$](http://latex.artofproblemsolving.com/e/5/d/e5d84c692bb0db11678ce289ae0ae6e1f6ba3d4a.png)
. Prove that 
is isosceles.
This post has been edited 1 time. Last edited by parmenides51, Aug 7, 2020, 9:22 PM
is cyclic and as 
is an isosceles trapezoid.
