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 open unit disk in the complex plane and let 
be a holomorphic function such that 
. Let the Taylor series of 
be 
. Prove that the number of zeroes of (counted with multiplicities) equals 
.
such that satisfies two following statements?
which contains 
.
if 
is an odd natural number, 
if is an even natural number ( is different from 0).
. Prove that for all polynomial 
on 
, we have
Can I change 1 by a constant 
?
be holomorphic on 
, continuous on 
and such that the limit 
exists. Prove that 
attains its maximum on the circle 
.
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
be triangle with point 
and 
on 
and 
are on the same side of 
be midpoint of segment and 
be midpoint of segment 
be intersection of 
with 
and 
be a convex polygon with centroid , and let 
be the set of vertices of . Let 
be the set of triangles with vertices all in . We sort the elements 
of into the following three types: lies in the strict interior of ; let 
be the set of triangles of this type. lies in the strict exterior of ; let 
be the set of triangles of this type. lies on the boundary of .
, denote by 
the area of . Prove that ![\[\sum_{T \in \mathcal A} S_T \geq \sum_{T \in \mathcal B} S_T.\]](http://latex.artofproblemsolving.com/5/e/d/5ed261e1e812466ea3e78f3b681f6bbdb7ebccd1.png)
be triangle with circumcenter 
and orthocenter . 
intersect again at 
respectively. Lines through parallel to 
intersects at 
respectively. Point such that 
is a parallelogram. Prove that lines 
and are concurrent at a point on 
.
![$f(x)\in\mathbb Z[x]$](http://latex.artofproblemsolving.com/2/9/5/2953d049559ec5d4020e6777d965bdfff9ebde0f.png)
have positive integer leading coefficient. Show that there exists infinte positive integer 
such that the number of digit that doesn'r equal to 
is no more than 
be a triangle and let be its circumcenter and its incenter. be the radical center of its three mixtilinears and let be the isogonal conjugate of . be the Gergonne point of the triangle .
is parallel with line 
.
has at least one pair of positive integer solution 
for each positive integer 
.
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
we have 
.
.
digit numbers are there
? and 
?
