AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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 
and 
such that 
is an obtuse angle. Let the circumcircle of 
intersect and respectively in points 
and 
. Let the line 
intersect in 
; let the line 
intersect in 
. Prove that, the points 
are concyclic.
such that 
.
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly times and there exist integers 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
, there exists an integer such that 
and 
are both divisible by . How many such labelings exist?
be an odd prime. Define 
over the positive integers as follows:
be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
denote the set of all positive integers less than 
that are relatively prime to . Let 
. Is the polynomial 
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 
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 is equal to the corresponding sum for the matrix . Prove that 
.
be an acute triangle and let be an interior point for it such that 
. Prove that
When does equality hold?
be a non-equilateral triangle, and let and be its incenter and circumcenter, respectively. Prove that 
if and only if 
, and that both inequalities become equalities simultaneously.
and 
.
