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 an acute triangle. Let 
be a point on side 
and 
be a point on side 
such that lines 
and 
are parallel. Let 
be an interior point of 
. Suppose rays 
and 
meet side at points 
and 
, respectively, such that both and lie between 
and 
. Suppose that the circumcircles of triangles 
and 
intersect at a point 
. Prove that the points 
, and 
are collinear.
. Prove that:
and 
, define 
as the greatest exponent 
such that 
divides 
. Prove that there are infinitely many such that 
and infinitely many 
such that 
.
with circumcircle 
is given. Let be the midpoint of , be the foot of altitude from 
to , and 
be the intersection point of and . Point 
lies on the arc of (the one that does not contain ) such that 
. Prove that 
.
be a trapezium inscribed in a circle 
with diameter . Let be the intersection point of the diagonals and 
. The circle with center and radius 
meets at the points 
and 
(where is on the same side of as ). The line perpendicular to at intersects 
at 
. Prove that 
is perpendicular to 
.![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
and 
. Prove that
Equality holds when 
or 
, , and lie on a circle such that 
. The tangent line to at point intersects the tangent line to at and the line 
at and . The circle and the circumcircle of triangle 
intersect again at . Prove that 
. and . Prove that
Equality holds when or 
intersect its circumcircle at 
. The tangets to from 
determine a triangle 
, where 
is relative to and so on. Prove that lines 
are concurrent at a point and that belongs to the euler line of . be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at , and 
intersect 
at . Prove that lines 
and 
are either parallel or coincide.
is a convex quadrilateral; half-lines 
and 
meet at point ; half-lines 
and meet at point . It is known that 
. The bisector of angle 
meets the sides and of the quadrilateral at points and , respectively; the bisector of angle 
meets the sides and at points 
and 
, respectively.
and 
meet at point inside the quadrilateral. lies on the diagonal .
is a convex quadrilateral; half-lines and meet at point ; half-lines and meet at point . It is known that . The bisector of angle meets the sides and of the quadrilateral at points and , respectively; the bisector of angle meets the sides and at points and , respectively. and meet at point inside the quadrilateral. lies on the diagonal .
is the radical axis of the circles 
and 
, therefore their intersection in the interior of the quadrilateral will trivially lie on . is their axis. We will prove only that 
, where 
is the radical axis, for 
beeing the same thing.
, (the power of 
w.r.t to the circles), but we have that 
, because the quadrilateral 
is cyclic, therefore we only need to prove that :
, which is obvious because the triangles 
and 
are similar.
and 
.
and 
.
Where 
