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 a polynomial 
with real coefficients of degree 
(
) such that has no real roots. Prove that the polynomial![\[
Q(x) \;=\; P(x) \;+\; a\,P'(x) \;+\; a^2\,P''(x) \;+\; \dots \;+\; a^n\,P^{(n)}(x)
\]](http://latex.artofproblemsolving.com/6/5/e/65e5e4eefcadd5d0019b4929b3472cdf203c13aa.png)
has no real roots.
of these points in such a way that any line 
that forms an angle of 
or 
with the positive horizontal semi-axis intersects at exactly 
points or at no points.
is called good, if 
are non-negative integers, and 
is either 
or 
for all 
.
, and Ivan has a whiteboard with some ones written on it. In each step, he may erase any good sequence that appears on the whiteboard, and then he writes the number 
such that 
Suppose Ivan starts with the least possible number of ones to obtain 
after some steps, determine the minimum number of steps he will need in order to do so.
, 
, 
cut three common chords concurrent at 
. Consider two distinct circles 
, 
which are internally tangent to all 
. Determine, with proof, which of the following two statements is true. is the insimilicenter of and is the exsimilicenter of and .
interesting if is composite and for every divisor 
of at least one of 
and 
is also a divisor of with 
be reals such that 
Prove that
Let be reals such that 
Prove that
Let be reals such that 
Prove that
satisfy that: 
and 
. Find the maximum: 
be non-negative real numbers. Prove that
and 
is tangent to the circumcircle of triangle through point 
. Points 
and 
are the midpoints of segments 
and 
respectively, then line 
and line 
intersect at points 
and 
respectively. Line 
and line 
intersect the circumcircle of triangle at points and 
respectively. Prove that , , and concyclic.
be a acute, non-isosceles triangle. 
are the midpoints of sides 
, resp. Denote by 
the circumcircle and Euler circle of . An arbitrary point 
lies inside triangle 
and 
intersect 
at 
, resp. Point 
is the point such that 
is the midpoint of 
. Points 
are defined similarly.
then 
; is mirrored by 
, its image is . 
are created in the same manner. 
is the orthocenter of and 
intersect 
at 
resp. Prove that are collinear.
and 
. Prove that
Equality holds when 
or 
or 
touches hypotenus 
at , and 
at 
and 
respectively. 
and 
are reflections of in 
and 
.
, where 
is the incenter of .
because 
and so 
- bisector of 
so 