students, each pair of students have watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
be non-negative real numbers such that 
Prove that
and 
.
Y by
Let 
be an acute triangle with 
, inscirbed in circle 
, with center 
. Circle 
, with center point 
and radius 
intersects 
at point 
and the circle at point 
. Line 
intersects circle at point 
. The circumscribed circle 
of triangle 
, intersects at point 
. Prove that:
a) Point
is the center of circle
b) Circumscribed circle of triangle
is tangent to .
be an acute triangle with 
, inscirbed in circle 
, with center 
. Circle 
, with center point 
and radius 
intersects 
at point 
and the circle at point 
. Line 
intersects circle at point 
. The circumscribed circle 
of triangle 
, intersects at point 
. Prove that:a) Point
is the center of circle b) Circumscribed circle of triangle
is tangent to .
and since 
, 
so 
so 
and 
which proves the tangency.
![\[\angle FDB=\angle ADB=\angle ADC=\angle ACD=\angle ACB=\angle DFB\]](http://latex.artofproblemsolving.com/b/8/1/b810fd5a04c4405975d95e9a025ad7bbd0b78f6b.png)
![\[\angle EBC=\angle EAC=\angle 2EDC\]](http://latex.artofproblemsolving.com/2/f/c/2fc300d278c723c2cf609fc861524d15da8d8353.png)
![\[\angle GEC=\angle DGE -\angle DCE=\angle DFE -\angle DCE=\angle ACE -\angle DCE=\angle DCA=\angle BCA\]](http://latex.artofproblemsolving.com/3/a/f/3af5adb66e6cad872132e2666b4892631f4a0d1c.png)
