AoPS Academy
is described on a circle 
, and is not a trapezius inscribed in a circle. Let the tangency points of and 
be 
respectively. A circle with a center 
, different from is tangent to the segement 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines . A circle with center 
, different from is tangent to segment 
and lines . Prove that the lines 
are concurrent.
such that 
is a divisor of 
.
, such that 
and 
.
such that 
is prime, 
with 
and ?
. He says to Cecilia: "If you pick any subset of 10 elements from , it will be disjoint with at least one subset from my list." What is the smallest possible number of subsets Miguel's list can have if what he tells Cecilia is true?
be the radius of the circle circumscribed to the triangle 
of sides 
, 
, and 
. If 
, find the angles of the triangle.
be the radius of the circle circumscribed to the triangle of sides , , and . If , find the angles of the triangle.
