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 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 
.
and 
such that
such that 
is the cube of a positive integer. Find .
with respect to 
is greater than 0 if lies between 
and 
. (I've underlined it). But, I'm a little confused as I thought the power was 
and since is inside the circle, wouldn't the power be negative since 
?
.
.