AoPS Academy
with right angles at 
and 
has an inscribed circle with center 
and radius 
. Let the lengths of the parallel sides 
and 
be 
and 
, and the intersection of the diagonals 
and 
be 
. to one of the trapezoid sides has the length . and as a function of and .
be an acute triangle with incenter 
and circumcenter 
. The incircle touches sides 
and 
at 
and 
respectively, and 
is the reflection of over . The circumcircles of 
and 
meet at 
, and the circumcircles of 
and meet at a point 
, where is the midpoint of 
. Prove that if 
and meet at 
, then 
.
such that divides 
but 
does not divide 
.
be a point on the Neuberg cubic. Show that as varies, the Nine Point Circle of the antipedal triangle of goes through a fixed point.
be distinct prime numbers. Let ![$\alpha_i=\sqrt{a}, \,\mathrm{A}=\mathbb{Q}[\alpha_1,\alpha_2,\hdots,\alpha_b]$](http://latex.artofproblemsolving.com/0/e/0/0e06919fe05d9f17760890bb2aebb9cfa85d54ce.png)
![$[\mathrm{A}:\mathbb{Q}]=2^b$](http://latex.artofproblemsolving.com/b/e/1/be12fe0e1becd8a9dcddd1eece210f2df8ecd026.png)
![$\mathrm{A}=\mathbb{Q}[\gamma];$](http://latex.artofproblemsolving.com/f/6/3/f639e791853133a444708bf47517457379d347e8.png)
and deduce that the minimum polynomial 
of 
over 
has degree 
factors in ![$\mathbb{Z}_a[X]$](http://latex.artofproblemsolving.com/e/9/9/e994e6e9083e52f44aef280b4346f6994a9d9f02.png)
into a product of polynomials of degree 
either of degree 
be distinct prime numbers. Let
and deduce that the minimum polynomial of over has degree factors in into a product of polynomials of degree either of degree 
