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.
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.
be the inscribed circle of a triangle 
. Let 
and be the tangency points of and the sides 
and , respectively, and let 
and 
intersect at 
and 
, respectively, such that 
and are all distinct. The tangent line of at 
intersects 
at 
, the tangent line of at 
intersects at 
, and the tangent line of at M intersects 
at 
. Prove that 
and are collinear.
be an acute triangle with 
. Denote by and points on the segment such that 
. 
is a point on segment . 
intersects 
and 
at 
and 
, respectively. The angle bisectors of 
and 
intersect at . If 
and 
, prove that 
is cyclic.
be a scalene triangle with incircle 
.
be the circle tangent to and passing through and , and denote by 
, 
the second intersection points of and , , resp. Define 
, 
, 
, 
cyclically.
be the circle internally tangent to 
, 
, 
. is tangent to 
.
inscribed in 
. Two points lie on such that 
are isogonal in 
. is the midpoint of 
. lies on 
such that 
. 
intersects at . Prove that the line through parallel to 
passes through the Euler center of triangle .
be an acute triangle. Let be the altitude on , and let be any interior point on . Lines 
, when extended, intersect 
at 
respectively. Prove that 
.
, 
the point 
is the midpoint of 
and 
is a point on the side such that 
. The cotangents of the angles 
, 
, and 
are positive integers 
respectively.
is also an integer and equals 
.
, some of which consisting of Fibonacci numbers.
be an acute triangle with incentre and 
. Let lines and intersect the circumcircle of at 
and 
, respectively. Consider points and such that 
and 
are parallelograms (with 
, and 
). Let be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
and 
be the projection of 
on the plane formed by 
If 
denotes the area of the region 
show that the following affirmations are equivalent:
