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.
, and the centroid of the triangle formed by the centers of the three smaller circles, B. What is the length 
.
, 
be such points on sides 
, 
of a triangle 
that 
is cyclic. 
, 
is the midpoint of 
. In triangles 
and 
let 
, 
be the 
-humpty points and let 
, 
be the -dumpty points. Prove that 
and 
.
and 
with 
. Determine the smallest positive integer 
satisfying the following condition:
, 
, 
, there exists a real-coefficient polynomial 
in variables satisfying the following properties: has degree at most ; is not identically 
;
of 
, the equation 
holds.
![\[
\sqrt{x-3}+\sqrt{6-x} \leq p
\]](http://latex.artofproblemsolving.com/3/2/b/32b8bf5c8aa4d7561933e38ee0e2a0480e1a2f07.png)
In which 
, hence find the largest value of 
.
there exists an -digit number divisible by 
all of whose digits are odd.
be non-zero distinct real numbers so that there exist functions 
so that:
and large enough 
.
so that:
and large enough .
be a quadrilateral with both an incircle and a circumcircle. Let 
and 
be the incenter and circumcenter of , respectively. Let 
be the intersection of lines and 
, and let 
be the intersection of lines 
and 
. Let and be the intersections of the line 
with lines and , respectively. Prove that the circumcircle of 
, the circumcircle of 
, and the line 
are concurrent.
such that 
.
. Find the minimum and maximum values of 
+ 
+ 
such that
for any 
,
into the expression, you might notice things simplify quite nicely 
.
,