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.
lies inside a triangle 
on the bisector of angle 
. Let 
and 
be the circles touching 
and 
at and passing through ; 
and 
be the common points of and with the circumcircle of distinct from . Prove that the circumcircles of the triangles 
and 
are tangent.
as 
, where 
are integers. Determine the number of ways to represent in this form and prove your conclusion.
such that for any positive integer 
, satisfies 
.
such that 
for all real numbers 
and 
be an integer. Let 
, 
, ..., 
be positive real numbers such that ![\[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\]](http://latex.artofproblemsolving.com/1/a/b/1ab410ebc17d9f6c38a3e7ae0f6efa955ffddfd3.png)
Show that 
, 
, 
are side lengths of a triangle for all 
, 
, 
with 
.
and , and 
positive (not neccesarily distinct) real numbers 
, 
. An equation is written on a whiteboard: 
where 
is a fixed positive real number, with exactly asterisks.
, while Rubi fills each asterisk with a number from 
, so that the equation on the whiteboard is correct. Suppose for every positive real number , the number of ways for Ebi and Rubi to do so are equal. and are permutations of each other.
and 
are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example 
.)
for all 
.
and 
with centres 
and 
are externally tangent at point 
and internally tangent to a circle 
at points 
and 
respectively. Line 
is the common tangent of and at . Let 
be the diameter of perpendicular to , so that 
are on the same side of . Prove that lines 
, 
, 
and are concurrent.
then 
,
.