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 the circumcenter, symmedian point of 
. Show that the tangent of 
at 
,the tangent of 
at 
, the tangent of 
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly positive integer factors 
. Assume that no prime larger than 
divides any of the 
's. Show that there must be some perfect cube among the 's.
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
, 
and 
.
, where 
are real numbers such that 
is an integral multiple of 
. Using (a) or otherwise show that if 
, then 
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
. The circle 
centered at the midpoint of 
and passing through intersects the sideline at points 
and 
. Similarly, define the points 
, 
, 
and 
., , , , and are concyclic.
with real coefficient. It is known that 
and 
. Prove that 
can't have 
distinct real roots.
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.
, then we know that 
.![[asy]
import olympiad;
import geometry;
size(250);
currentpen=fontsize(9);
pair a,b,c,h,a1,b1,a2,b2;
a=dir(105);
b=dir(-90-50);
c=dir(-90+50);
h=a+b+c;
a1=foot(a,b,c);
b1=foot(b,a,c);
a2=foot(a1,b,b1);
b2=foot(b1,a,a1);
draw(a--b--c--a--a1--a2^^b--b1--b2--a2);
label("$A$",a,dir(a));
label("$B$",b,dir(b));
label("$C$",c,dir(c));
label("$H$",h,dir(-30));
label("$A_1$",a1,dir(a1));
label("$A_2$",a2,dir(180-40));
label("$B_1$",b1,dir(b1));
label("$B_2$",b2,dir(b2));
[/asy]](http://latex.artofproblemsolving.com/f/f/5/ff59bc10a58d8f310a75cf547e536530346c12f4.png)
, 
is an altitude, so 
. This implies that 
, or that 
. Similarly, 
. In order to show that , we will show that 
. By virtue of our previous calculations, we know that
On the other hand, since 
is cyclic, by Power of a Point we know that 
. Thus,
which implies that as desired.
is cyclic since 
. It is well known that so is 
. So by angle chasing we have,
and 
.
and 
be heights in acute triangle intersects at 
be heights in triangles 
and 
,
and 
are antiparallel in 
it suffices to show that 
so 
is cyclic. 
.