[/quote]
,
such that 
.
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 
.
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that 
is an 
matrix with entries 
,
such that the sum of the elements in each row and each column of 
.
be an interior point for it such that 
.
When does equality hold?
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
have the same incenter 
.
is concyclic iff 
is concurrent
be the nonnegative integers in 
where 
.
.
.
such that 
where 
,
?
where 
,
be a scalene triangle with circumcircle 
and circumcircle 
.
and 
as the midpoints of 
and 
,
to 
.
meets 
,
intersects 
.
at 
and 
at 
.
intersect 
,
intersect 
.
is the reflection of 
and 
lies on the circumcircle of 
.
and 
and determine the equality cases.
Y by Adventure10, Mango247
A right triangle 
with hypotenuse 
is inscribed in a circle. Let 
be the midpoint of the arc 
not containing 
the midpoint of side 
, and 
a point of intersection of ray 
with the circle. Let 
be a point of intersection of tangents to the circle at points 
and 
. Prove that 
.
with hypotenuse 
is inscribed in a circle. Let 
be the midpoint of the arc 
not containing 
the midpoint of side 
, and 
a point of intersection of ray 
with the circle. Let 
be a point of intersection of tangents to the circle at points 
and 
. Prove that 
.
is a symmedian of 
.
so ![\[\angle PKN=\angle AKC-\angle AKP-\angle CKN=90^{\circ}+\frac12(BC)-\frac12(CM)-\frac12(CM)=90^{\circ}\]](http://latex.artofproblemsolving.com/9/4/6/946bc5a9708cb0cb8120aca0ce311be32f6fb7a8.png)
as desired.
