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 a triangle with incenter 
. The foot of the perpendicular from to 
is 
, and the foot of the perpendicular from to 
is 
. Prove that 
. 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 an acute triangle. Let be the altitude on , and let 
be any interior point on . Lines 
, when extended, intersect 
at 
respectively. Prove that 
.
with 
, let be the bisector of with on the side . Let 
be a circle passing through 
and which is tangent to at . Suppose cuts the side 
again at 
. The tangent to the circumcircle of 
at intersects again at 
. Let be the intersection point of the segments 
and . Prove that 
is perpendicular to .
and 
are natural numbers such that 
and 
and 
(by 
we mean a number from the set 
which is congruent with 
modulo ). if for 
.( the symbol 
means the number of elements of the set)(
points)
such that ![\[p^q+q^p+p+q-5pq\]](http://latex.artofproblemsolving.com/b/a/b/bab8a0ee3555cf1136b6de56179012b920d0a2ad.png)
is a perfect square.
satisfying![\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]](http://latex.artofproblemsolving.com/a/e/6/ae65e3d53931fe9d60d0de76dc79540ccf44c26e.png)
and let 
denote the largest real number satisfying 
for all positive integers . What are the possible values of across all possible choices of the sequence 
? with circumcircle , and two arbitrary points 
on . Let , 
, 
be points on lines , 
, , respectively, such that , 
, and 
concur at a point . Let 
be a point on line such that 
, 
, , are concyclic. Similarly, let 
be a point on line such that , , , are concyclic, and let 
be a point on line such that , , , are concyclic. Prove that 
, 
, 
concur at a single point.
such that 
is a square of an integer.
be a positive integer. Prove that there exists a poisitve integer 
such that
and 
Prove that
,
,
and 
of the sides of the quadrilateral satisfy the inequalities ![\[ 2\le a^2+b^2+c^2+d^2\le 4. \]](http://latex.artofproblemsolving.com/c/c/9/cc9b5e4ea15ad064d9fc812efe0db96fd1419905.png)
,
,
,
with 
(draw a diagram and this will be more obvious; it's just the Pythagorean Theorem). Then
.
,
.