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.
such that:
be a triangle with circumcenter 
. Let 
be the midpoints of 
and 
respectively and let 
be the projection of on 
. Let 
be the projection of 
on 
. Let 
intersect 
at points 
and 
. Let 
intersct at points 
. Let 
intersect 
at 
and 
intersect at 
. Then show that 
are concurrent.
and 
respectively be the incentre, circumcentre and circumcircle of triangle 
. Points 
are chosen on , such that 
, and point 
is the foot of the altitude from 
to 
. If 
are similarly defined, prove that lines 
and 
concurr on 
.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are distinct positive integers and 
for every positive integer 
. The terms of the sequence are not known and the goal is to find the period . To do this, at each move it possible to reveal the value of a term of the sequence at your choice. is one of the first 
prime numbers, find for which values of there exist a strategy that allows to find revealing at most 
terms of the sequence.
such that, for all real numbers 
with 
, 
of positive integers for which there exist positive integers 
and 
such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
number. I will allow you to ask me questions. A question consists of you saying a three-digit number, to which I respond 'yes' if at least two of its digits are correct, or 'no' otherwise. (A digit is correct if it matches the corresponding digit in my number.)
on the diagonal 
of a given rectangle 
, such that 
. The point is the intersection point between 
and the parallel line to 
that contains 
. Prove that the triangle 
is equilateral if and only if is a square.![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
, 
is an homoemorphisme too.
checkered board is cut into several squares with integer side length. The size of a square is it's side length.
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.
. Also 
. We have 
. We will proceed by using induction. Suppose that 
holds, we will prove 
. On the other hand, for all 
Thus, we obtain the following
implying that the sequence 
must be eventually constant as desired.
. Also . We have . We will proceed by using induction. Suppose that holds, we will prove . On the other hand, for all Thus, we obtain the followingimplying that the sequence must be eventually constant as desired.
Prove that
and let 
and 
be tangency points of the incircle of 
,
and 
respectively. Let the circle with centre 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
s
