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.
be the intersection of 
and . It is well-known that in configurations like this one, the midpoint of arc 
on 
is the center of the spiral similarity mapping to 
, and that the line 
is parallel to the -angle bisector by this spiral similarity. Next, see that the -angle bisector is parallel to the internal angle bisector of 
and we are done.
the midpoint of the line segment . Let 
. Notice that 
, and 
.
is a parallelogram. This gives 
and 
.
and 
.
Where 
.
is perpendicular to line which connects middles of sides of line segments 
and 
![[asy]
size(200);
defaultpen(fontsize(8pt));
pair A = (0,0);
pair B = (1,2);
pair C = (4,2);
pair D = (3,0);
pair P = rotate(15,C)*B;
pair M = midpoint(A--P);
pair N = midpoint(C--D);
pair Pp = extension(B,P,M,N);
pair I = midpoint(P--B);
draw(A--B--C--D--cycle^^C--P, lightblue);
draw(M--N, orange);
draw(B--P, magenta);
draw(P--Pp, dashed+magenta);
draw(A--P, palered);
draw(M--I--C, dashed+purple);
dot("$A$", A, dir(-100));
dot("$B$", B, dir(120));
dot("$C$", C, dir(40));
dot("$D$", D, dir(-30));
dot("$P$", P, dir(135));
dot("$M$", M, dir(140));
dot("$N$", N, dir(N));
dot("$I$", I, dir(I));
[/asy]](http://latex.artofproblemsolving.com/f/a/1/fa100a270c6133ab0a4e56e01a6efe6494e1067b.png)
be the midpoint of 
be the midpoint of 
is a parallelogram.
and 
since 
is the midline of 
.
.
.
,
is an isoceles triangle, we have 
.
,
,
,
.
,
,
,
.
.