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 triangle with circumcenter 
and internal bisector 
. Let cuts 
again at 
and 
cuts again at 
. Point 
lie on such that 
. The line pass through and perpendicular to intersects 
at 
respectively. Let circumcircle of 
cuts at 
. Prove that 
.
eggs, each with a hardness value 
, where 
is a permutation of the set 
. He is throwing the eggs from an 
-floor building.
-th egg is dropped from the 
-th floor, its new hardness becomes![\[
\left\lfloor \frac{k_i}{j+1} \right\rfloor.
\]](http://latex.artofproblemsolving.com/7/5/8/7583af9655a43eb3f6167f955148984a468d4e70.png)
If 
, then the egg breaks and cannot be used again. and can Arthur always determine the correct ordering of the eggs according to their initial hardness values?
. The interior bisector of 
intersects 
at 
and the minor arc of 
in circumcircle of at . Suppose that 
is a point on the minor arc of such that 
. 
is a point on the line segment of such that 
. is the circumcenter of . Prove that 
.
be reals such that 
and 
Find the value of 
be a triangle with circumcentre . The points 
lie in the interiors of the sides 
respectively, such that 
is perpendicular to 
and 
is perpendicular to 
. (By interior we mean, for example, that the point lies on the line and is between 
and 
on that line.)
be the circumcentre of triangle 
. Prove that the lines 
and are perpendicular.
, a line 
to tangent , and another circle 
disjoint from such that and lie on opposite sides of . The tangents to from a variable point 
on meet at 
and 
. Prove that, as varies over , the circumcircle of 
is tangent to two fixed circles.
and we define 
as the set of all products of an element of and an element of . For example, if 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and are the same. Prove that the set of perfect powers greater than or equal to 
is good., , no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form 
, where 
and 
are integers.) be an acute triangle. Let be the midpoint of side , and let and 
be the feet of the altitudes from and , respectively. Suppose that the common external tangents to the circumcircles of triangles 
and 
intersect at a point , and that lies on the circumcircle of . Prove that line 
is perpendicular to line .
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
be the number of positive integers that divide the integer . For all positive integral divisors 
of 
, what is the sum of numbers 
?
.
.
.