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.
has circumcircle 
and circumcenter 
. A circle 
with center 
intersects the segment 
at points 
and 
, such that 
, , , and 
are all different and lie on line in this order. Let 
and 
be the points of intersection of and , such that , , , , and lie on in this order. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point 
. Prove that lies on the line 
.
of positive integers such that 
is an integer.
be an acute triangle, 
its orthocenter and the center of its nine point circle. Let 
be a point on the parallel through to such that 
and and are on different sides of and 
a point on the parallel through to 
such that 
and and are on different sides of . If 
and 
are the reflections of over and respectively, 
and 
are the intersections of 
and 
respectively with the circumcircle of , prove that the intersection of lines 
and 
lies on 
.
undergoes the following procedure in order to obtain the number 
: to the first position to obtain the numb er 
; to obtain the number 
; to the end to obtain the number 
.
.) For example, for 
, we get 
, 
, and 
.) for which 
.
show that there exists a prime 
such that one can choose distinct integers 
such that p divides 
for all 
.
and a positive integer 
is given.
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
is a pentagon whose vertices lie on circle 
where 
. Let 
and intersect at , 
meet 
at . 
is the midpoint of arc on , not containing . If 
holds, then what is the value of 
?
and such that 
and 
is a prime number
be an acute triangle and its orthocenter. Let be a point on the parallel through to such that 
. Define and 
as points on the parallels through to and through to similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be an acute triangle with 
, its orthocenter and it's circumcenter. Let be a point on the parallel through to such that 
and and are on different sides of . Denote by the intersection of the circumcircle of and 
, where 
is the reflection of over , the midpoint of 
, the intersection of 
and the parallel through to 
, the intersection of 
and the perpendicular through to 
and 
a point on such that 
, where is the midpoint of 
. Prove that 
lie on a line.
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both and . Integers and are coprime if 
., which doesn't have right angle, let be its circumcenter. Circle 
intersects and at 
and 
for the second time, respectively. Similarly, circle 
intersects and 
at 
and 
, and circle 
intersects and 
at 
and 
for the second time, respectively. Let 
and 
be circumcenters of triangle 
and 
, respectively. Prove that 
are collinear.
of integers is rootiful if, for any positive integer 
and any 
, all integer roots of the polynomial 
are also in . Find all rootiful sets of integers that contain all numbers of the form 
for positive integers and .
be the midpoint of 
. Drop perpendicular 
to 
. right triangle 
has 
So 
. So 
.
be the respective midpoints of sides 
of triangle 
.
centered at 
.
gets mapped to 
, 
to 
and 
to .
.
. So 
concyclic. So lies on the circumcircle of 
which is the medial triangle of 
.. belongs to the incircle as well as nine-point circle of .
.
is point of contact of Incircle And Nine-Point Circle, Then 
,
being inradius.
are conciclic
.
and 
.
be the tangency point of the incircle with 
,
is equilateral. So 
.
.