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.
such that 
for all 
.
such that the equation
such that 
. take a random 
on the altitude from 
to 
.
intersects second time in 
. Take 
such that it's on the segment and 
and 
.The second intersection of 
and circle 
is 
, (
) and the second intersection of 
and circle 
is 
,(
).The tangent from 
to the circle intersects the altitude from at 
.
is tangent to circle 
.
are 
and 
Find the product of all possible values of 
be a set consisting of 
positive integers, none of which is a sum of two other distinct members of . Prove that the elements of may be ordered as 
so that 
does not divide 
for all 
.
be an acute triangle with 
and 
. Let be the midpoint of the side 
. The circumcircle of triangle 
intersects the side again at 
and the circumcircle of triangle 
intersects the side again at 
. The point lies on the perpendicular bisector of the segment so that the points and lie on the same side of 
and 
. Prove that the circumcircles of triangles 
and 
are tangent.
![\[ p = \frac{2ab}{a + b} \]](http://latex.artofproblemsolving.com/0/f/4/0f4297ec59fe97c92fda75af4cc835be6b9a2507.png)
, ![\[ p, a, b \in \mathbb{C} \]](http://latex.artofproblemsolving.com/b/4/4/b4499970ac49d770a810d7345fedef7773cd03f5.png)
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
has rational sides and equal angles. Show that it is regular.
and of an equilateral triangle with side 
, points and 
are taken, respectively, such that the segment 
touches the circle inscribed in the triangle and its length is 
. Find the area of the triangle 
.
and of an equilateral triangle with side , points and are taken, respectively, such that the segment touches the circle inscribed in the triangle and its length is . Find the area of the triangle .
and 
.
Prove that
Prove that
Let 
Prove that
![\[
AP+AQ=2AB-CQ-BP=2AB-AB-PQ=AB-PQ=10-4=6
\]](http://latex.artofproblemsolving.com/3/4/7/347bc101441bce8fbc693313e57079c58778a679.png)
So 
.