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.
are there such that 
and 
?
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of 
less than is the difference between two numbers on the sheet, and no other positive number is.
be a square with 
side length and with center 
. A rhombus 
is drawn where point 
lies on line 
. What is the area of ?
numbers 
, calculates the list 
where 
is the number of times the number 
appears in the list. So, for example, for the list 
, the program returns the list 
.
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of for which, whatever the initial list of positive integers 
, it is possible for Alexander to do what the teacher asked him to do.
table is colored black or white. Every cell on table border is black. It is known, that in every 
square there are cells of two colors. Prove, that exist square that is colored in chess order.
be a non-isosceles triangle with orthocenter 
, circumcenter 
. Let 
be midpoints of 
and 
be reflections of across 
. Prove that triangle 
have the same centroid.
be a quadrilateral such that 
and 
are acute and 
. Suppose that 
, prove that 
.
. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was 
, what is the fifth term of the sequence?
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
be circumscribed about circle 
, and let be the orthocenter of 
. The circle touches line 
at 
. The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at . Prove that 
is tangent to .
![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
of triangle lies a point with the following properties:
is perpendicular to the bisector of 
.
.
and 
labeled . We then label 
as 
. It then follows that 
. Next, label 
as 
. Then, 
and 
also equal . Noting 
, we can now write the following equations,
Subtracting the bottom equation from the top one gives 
. We can now write the equation 
and solving gives 
. Thus, we now know all the angles of , specifically, 
and 
and natural number 
and 
.
