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 right angled triangle with 
.Let 
be the midpoint of 
, and 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
define 
if 
or 
, and 
where 
are all prime factors of . Find all polynomials 
with nonnegative integer coefficients such that 
divides 
for every nonnegative integer .
satisfy the equation 
. Prove that there exist positive integers 
such that
be a given integer. Prove that infinitely many terms of the sequence 
, defined by ![\[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\]](http://latex.artofproblemsolving.com/8/3/2/832113b1ef1fe211e1eaead25129873aead1dab4.png)
are odd. (For a real number 
, 
denotes the largest integer not exceeding .) be a triangle with incentre 
and circumcentre 
. Let 
be the touchpoints of the incircle with , 
, respectively. Prove that 
is the Euler line of 
.
with its incircle 
. Let 
and be the points of tangency of with and respectively. On the edges 
and 
, take points 
and 
such that 
is tangent to at . Suppose is the intersection point of the lines 
and 
. Prove that two lines 
and 
are parallel or coincide.
be a triangle with orthocenter 
and let be the midpoint of 
Suppose that and are distinct points on the circle with diameter 
different from 
such that lies on line 
Prove that the orthocenter of 
lies on the circumcircle of 
be a scalene oblique triangle with circumcenter and orthocenter , and (
, 
) a point in the plane.
, 
be the circumcevian triangles of , , respectively.
be the pedal triangle of with respect to .
be the reflection in 
of 
. Define 
, 
cyclically.
be the reflection in of 
. Define 
, 
cyclically.
, 
be the circumcenters of 
, 
, respectively., , are collinear if and only if lies on the Jerabek hyperbola of ., , are collinear if and only if lies on the Lemoine cubic (= 
) of .
is a triangle with incenter . The foot of the perpendicular from to is 
, and the foot of the perpendicular from to 
is . Prove that 
.
with an obtuse angle 
, let 
be feet of altitudes from 
on sides 
respectively. The tangents from to circumcircle of triangle intersect line 
at points 
respectively and we know that 
. Point 
lies on segment 
in such a way that 
and let 
be a point on line such that 
is between 
and 
. Prove that the circle with diameter 
is tangent to circumcircle of triangle . be a triangle with 
. A circle 
is tangent to sides and , and is the midpoint of . Points and lie on sides and , respectively, such that segment is tangent to circle at point . Let and be the orthocenters of triangles 
and 
, respectively. Prove that line 
bisects segment 
. be a triangle with points and lying on rays and , respectively, such that 
. On the line , choose points and 
such that 
and 
. Let and 
be the feet of the perpendiculars from and to line , respectively. Let 
be the intersection point of lines 
and 
. Prove that the reflection of over line lies on the line connecting 
and the circumcenter of triangle .
and of an equilateral triangle , points and are chosen such that 
. Find 
, if is the intersection point of the segments 
and 
.
and of an equilateral triangle , points and are chosen such that . Find , if is the intersection point of the segments and .
