[/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 
matrix with entries 
,
such that the sum of the elements in each row and each column of 
.
be an acute triangle and let 
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
Inside a circle there are two intersecting circles. One of them touches the big circle in point 
, the other in point 
. Prove that if segment 
meets the smaller circles at one of their common points (fig.), then the sum of their radii equals the radius of the big circle. Is the converse true?
(A. Vesyolov)
, the other in point 
. Prove that if segment 
meets the smaller circles at one of their common points (fig.), then the sum of their radii equals the radius of the big circle. Is the converse true?
(A. Vesyolov)
This post has been edited 2 times. Last edited by parmenides51, Aug 21, 2019, 1:07 AM
,
and 
(which passes through 
and 
.
and 
are both isoceles, and have a common angle which is 
,
.
.
.
