[/quote]
,
such that:
Prove that
be a triangle with circumcenter 
be the midpoints of 
and 
respectively and let 
be the projection of 
.
on 
.
intersect 
at points 
and 
.
intersct 
.
intersect 
at 
intersect 
.
are concurrent.
and 
respectively be the incentre, circumcentre and circumcircle of triangle 
are chosen on 
,
is the foot of the altitude from 
.
are similarly defined, prove that lines 
and 
concurr on 
.
and 
and 
respectively. Let the circle with centre 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are 
for every positive integer 
.
prime numbers, find for which values of 
terms of the sequence.
such that, for all real numbers 
with 
,
s
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 
,
.
and the parallel line to 
that contains 
.
is equilateral if and only if 
![$P \in \mathbb R[x,y]$](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.
Y by
In a triangle 
, 
is the orthogonal projection of the incenter 
onto 
. Line 
meets the incircle again at 
. Line 
intersects side at point 
. Suppose that the segment IO is parallel to , where 
is the circumcenter of 
. If 
is the circumradius and 
the inradius of the triangle, prove that 
.
, 
is the orthogonal projection of the incenter 
onto 
. Line 
meets the incircle again at 
. Line 
intersects side at point 
. Suppose that the segment IO is parallel to , where 
is the circumcenter of 
. If 
is the circumradius and 
the inradius of the triangle, prove that 
.
