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 an even integer and let 
be real numbers satisfying 
.
such that 
, 
, 
, 
be four points such that no three are collinear and is not the orthocenter of 
. Let 
, 
, 
be the orthocenters of 
, 
, 
, respectively. Suppose that the lines 
, 
, 
are pairwise distinct and are concurrent. Show that the four points , , , lie on a circle.
and 
be the circumcenter and orthocenter, respectively, of an acute scalene triangle . The perpendicular bisector of 
intersects 
and 
at 
and 
respectively. Let 
denote the intersection of the circumcircles of triangles 
and 
other than .
and 
analogously by repeating this construction two more times. Prove that , , , and are concyclic.
be a quadrilateral inscribed in a circle 
. Let 
, 
, 
, 
, 
, 
, 
, 
. Prove that 
lies over 
.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist. be an odd prime and 
be integers so that the integers 
are divisible by . divides each of .
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible. be a triangle (
). 
are the altitudes of the triangle. The bisector of 
intersects 
at 
. Let be a point such that 
and 
. Prove that passes through the midpoint of 
.
is circumcircle of an acute triangle . is midpoint of arc 
and 
is incenter of triangle . Let 
intersect in 
and for second time in 
. Let be a point on line 
such that 
is parallel to 
. Prove that is bisector of angle 
.
.
power of 
and 
w.r.t circumcircle of 
is equal, so 
so 
and that means the perpendicular bisector of passes through which is fixed. So we are done.
) w.r.t a circle with radiius 
is equal to 
.
hits that circle and point and (assume that lies between and ) then 
. That means if we draw any chord 
of the circle passes through , we have 
.
.
and 
respectively, such that both 
