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 the circumcenter of an acute-angled triangle 
with 
. The line 
meets the side 
at 
. The circumcenters of the triangles 
and 
are 
and 
, respectively. Extend the sides 
and 
beyond 
, and choose on the respective extensions points 
and 
such that 
and 
. Prove that the quadrilateral 
is a rectangle if and only if 
.
satisfying
Prove
is a positive integer. Call all positive divisors of which are different from 
and beautiful divisors.We call a special number when it has at least 
beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.
that verify the following equation for all 
:
be a triangle with incenter 
and circumcircle 
. Circles 
passing through 
and 
passing through 
are tangent at . Let meet minor arc 
of at 
and at 
, and let meet minor arc 
of at 
and at 
. Rays 
and 
meet at 
. Let 
be a point such that 
is tangent to and 
is tangent to .
are collinear.
be an acute triangle and let be an interior point for it such that 
. Prove that
When does equality hold?
denote the set of all positive integers less than 
that are relatively prime to . Let 
. Is the polynomial 
reducible over the rational numbers, given that it has integer coefficients?
blocks, each weighing at least ; their total weight is 
. Prove that for every real number 
with 
you can choose a subset of the blocks whose total weight is at least but at most 
.
. Let be a subset of such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
is twice its height, and the height of rectangle 
is twice its width. The point lies on the diagonal 
. Which fraction of the area of the big rectangle is that of the small one?
is twice its height, and the height of rectangle is twice its width. The point lies on the diagonal . Which fraction of the area of the big rectangle is that of the small one?
is twice its height, and the height of rectangle is twice its width. The point lies on the diagonal . Which fraction of the area of the big rectangle is that of the small one?
.
.
and 
.
,
.
.
while 
together we get 