[/quote]
,
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
and 
be two circles with respective centres 
and 
intersecting in two distinct points 
is an obtuse angle. Let the circumcircle of 
intersect 
and 
.
intersect 
intersect 
.
are concyclic.
such that 
.
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
,
and 
are both divisible by 
be an odd prime. Define 
over the positive integers as follows:
where 
such that 
.
such that 
and 
is a perfect square.
denote the set of all positive integers less than 
that are relatively prime to 
.
reducible over the rational numbers, given that it has integer coefficients?
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 
.
.
When does equality hold?
Y by
An isosceles right-angled triangle 
is given. On the extensions of sides 
and 
, behind vertices 
and 
equal segments 
and 
were laid. 
and F are the points of intersection of the segment 
and the lines perpendicular to the 
, passing through the points and 
, respectively. Prove that 
.
is given. On the extensions of sides 
and 
, behind vertices 
and 
equal segments 
and 
were laid. 
and F are the points of intersection of the segment 
and the lines perpendicular to the 
, passing through the points and 
, respectively. Prove that 
.
