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 triangle with circumcenter 
and orthocenter 
. 
the altitudes of 
. A point 
lies on line 
such that 
. A point 
lies on the circumcircle of such that 
are concurrent. 
meets 
at 
. Prove that 
are concyclic.
be a triangle. Distinct points 
, 
, 
lie on sides 
, 
, and 
, respectively, such that quadrilaterals 
and 
are cyclic. Line 
meets the circumcircle of again at 
. Let 
denote the reflection of across . Show that lies on the circumcircle of 
. be a triangle with 
, and its shortest side. Let be the orthocenter of . Let 
be the circle with center 
and radius 
. Let be the second point where the line 
meets . Let be the second point where meets the circumcircle of the triangle 
. Let be the intersection point of the lines 
and 
.
is tangent to the circumcircle of the triangle 
.
with centroid 
and circumcircle 
centred at , the extension of 
meets at 
; lines and 
intersect at ; and lines and 
intersect at . Suppose the circumcentre 
of the triangle 
lies on and 
are collinear. Prove that 
.
be an acute triangle with 
the feet of the altitudes lying on 
respectively. One of the intersection points of the line and the circumcircle is 
The lines 
and 
meet at point 
Prove that 
is composite. 
- all divisors of . It is known, that 
are all divisors for some 
(except 
). Find all such .
be a positive integer, with divisors 
. Prove that 
is always less than 
, and determine when it is a divisor of .
we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
is tangent to the circle 
at . and cut the circle at and respectively. 
is the in-centre of , and is on the line 
. 
and intersect at , while 
and 
intersect at . Prove that the points 
lie on a circle.
be an acute triangle and its orthocenter. Let be a point on the parallel through 
to such that 
. Define and as points on the parallels through to and through 
to similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be an acute triangle, its orthocenter and the center of its nine point circle. Let be a point on the parallel through to such that 
and and are on different sides of and a point on the parallel through to such that 
and and are on different sides of . If 
and 
are the reflections of over and respectively, and are the intersections of 
and 
respectively with the circumcircle of , prove that the intersection of lines 
and 
lies on 
.
, is the angle bisector of 
. The circumcircle of intersects 
in 
respectively and the line through parallel to cuts 
in 
respectively. Prove that point 
lie on a circle.
kids in a class. Is it true that for all positive integers 
it is possible to create several clubs each with 3 kids such that any pair of kids are both present in exactly one club?
and 
be the midpoint of 
and 
the vertices of go to the respective midpoints
be the center of 
then 
be the touchpoints of the incircle with 
is the Euler line of 
.
the unit circle. Then 
.
,
.
are collinear 
.
,